Green Belt Case study
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Table of Contents Week 1
Classic Wastes ................................................................................................6
Prioritization Matrix .........................................................................................9
Week 2
Brainstorming ...............................................................................................10
Cause and Effect Diagram ..............................................................................12
Affinity Diagram ............................................................................................15
Tree Diagram ................................................................................................18
Interrelationship Diagram ...............................................................................19
Process Decision Program Charts (PDPC) .........................................................20
Activity Network Diagram ...............................................................................21
Week 3
Project Charter: Description ............................................................................25
Project Charter: Blank Worksheet ....................................................................26
Project Charter: Example ................................................................................27
Project Charter Highlights ..............................................................................29
Critical-to-Quality Tree ...................................................................................34
Flow Chart ...................................................................................................35
Swim Lane Flow Chart...................................................................................38
Week 4
Basic Statistics Using the Texas Instruments TI-30 Calculator ............................40
Standard Deviation: 6 Steps to Calculation .......................................................45
Standard Deviation: Description and Example ...................................................47
Histograms ...................................................................................................48
Cell Intervals: Impact on Shape ......................................................................49
Cell Intervals: Rules of Thumb ........................................................................50
Stem-and-Leaf Plots ......................................................................................52
Box and Whiskers Plot ...................................................................................54
Check Sheets: Defect Location Check Sheet .....................................................56
Run Chart: Basic Construction ........................................................................57
Run Chart: Shifts ..........................................................................................58
Run Chart: Trends .........................................................................................59
Week 5
Target Values: Illustration 1 ............................................................................60
Target Values: Illustration 2 ............................................................................61
Target Values: Illustration 3 ............................................................................62
DPU, DPMO, PPM, and RTY ..........................................................................63
01/31/2017
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DPMO to Sigma Level Relationship .................................................................65
Pp and Ppk: Formulas ...................................................................................66
Pp and Ppk: Example ....................................................................................67
Week 6
Finding a Z Score: Example 1 .........................................................................68
Finding a Z Score: Example 2 .........................................................................69
Finding a Z Score: Example 3 .........................................................................70
Z Table: Form 1 ............................................................................................72
Z Table: Form 2 ............................................................................................75
Z Table: Form 3 ............................................................................................77
Type I and Type II Errors in Hypothesis Testing .................................................79
P-Values: Example 1 ......................................................................................80
P-Values: Example 2 ......................................................................................81
P-Values: Example 3 ......................................................................................82
Student's t Distribution ..................................................................................83
Student's t Distribution: Example ....................................................................84
Student's t Distribution Table ..........................................................................85
Paired t Test: Description ...............................................................................87
Paired t Test: Student's t Distribution Table .......................................................90
x-bar and R Control Chart Construction Rules ...................................................91
x-bar and R Chart: Example ...........................................................................92
Measurement Discrimination ..........................................................................93
Measurement Discrimination: Why Six Points?..................................................95
Rational Subgroup .........................................................................................98
Control Chart Construction: Formulas for Centerlines .........................................99
Control Chart Construction: Formulas for Control Limits ...................................100
Control Chart Tests: Shewhart's Test for Outliers .............................................101
Control Chart Tests: Shewhart's Test for a Sudden and Drastic Shift in Level ......102
Control Chart Tests: Shewhart's Test for a Trend/Gradual Change in Level...........103
Week 7
Failure Mode and Effects Analysis (FMEA)......................................................104
Poka-Yoke: Part 1 ........................................................................................105
Poka-Yoke: Part 2 ........................................................................................106
Poka-Yoke: Part 3 ........................................................................................107
Chi-Square: Example ...................................................................................108
Chi-Square: Table ........................................................................................109
Chi-Square Distribution: ν = 1 .....................................................................110
Chi-Square Distribution: ν = 5 .....................................................................111
Chi-Square Distribution: ν = 10 ...................................................................112
Chi-Square Distribution: ν = 15 ...................................................................113
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F Distribution .............................................................................................114
F Test for Variance: Example .........................................................................115
ANOVA – One Way ......................................................................................116
Analysis of Variance (ANOVA): One-Way Manufacturing Example ......................117
Analysis of Variance (ANOVA): One-Way Service Example ................................120
Design of Experiments (DOE) Terminology: Designed Experiment ......................123
Design of Experiments (DOE) Terminology: Response Variable ..........................126
Design of Experiments (DOE) Terminology: Factors and Levels ..........................127
Week 8
Control Plan ...............................................................................................128
Standard Operating Procedures .....................................................................129
Cp and Cpk: Formulas .................................................................................131
Cp and Cpk: Example ..................................................................................132
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Classic Wastes
Classic Wastes As a result of the evolution of the principles of the Toyota Production System into what is sometimes referred
to as “lean thinking,” a list of Classic Wastes has emerged (beginning with the work of Taiichi Ohno). Lean
thinking considers waste anything that adds cost without adding value. You might hear that lean practitioners
refer to the seven wastes instead of the eight wastes. Their list is as follows:
• Transportation
• Inventory
• Motion
• Waiting
• Overprocessing
• Overproducing
• Defects
At Villanova University we suggest eight classic wastes instead of seven simply because it spells out the easy
to remember acronym “DOWNTIME” and it takes into account the wasted non-utilized talent.
Example: The Classic Waste categories are: Defects, Overproduction, Waiting, Non-utilized talent,
Transportation, Inventory, Motion, and Extra Processing. Below is a list of the categories with four examples of
each.
Defects
• Keys that do not function on a new calculator
• Incorrect account number on bank checks
• Ink streaks on printed packaging
• Misspelled names on customer mailing lists
Overproduction
• Production quotas of a cell phone that exceed
customer demand
• More graduates of a certain degree than the
market requires
• A machine center production rate that exceeds
downstream demand
• Printing more copies of a magazine than will be
sold
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Classic Wastes
Waiting
• Idle downstream machine centers due to
unavailability of material
• Production lines shutting down due to missed
delivery by supplier
• Delay of construction as a result of late building
permits
• Customer delays in check-out lines as a result of
too few cashiers
Nonutilized Talent
• Inflated job requirements that specify an MBA
when the work is easily accomplished without
an advanced degree
• Specifying work content for a given process that
under-utilizes the labor skill available
• A supervisor with too few employees for optimal
span of control
• Requiring training of an employee that is
essentially redundant
Transportation
• Poor workflow requiring unnecessary movement
of material
• Disorganized office with inconvenient access to
files
• Poor layout for finished goods requiring too
much movement
• Excessive work-in-process requiring unnecessary
material movement
Inventory
• Product on hand that exceeds customer demand
• Ordering excessive quantities of sales brochures
• Excessive work-in-process for the machine
production rate
• Ordering too Imuch food for the company party
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Motion
• More steps than are necessary to complete an
assembly
• Too many steps in a measurement protocol for a
lab test
• Too many signatures required to approve an
expense report
• Inefficient layout for casting a part so that it is
lifted by the operator too often
Extra Processing
• Polishing a table that will be covered with a
tablecloth
• Printing in color when black and white is
acceptable
• Applying three coats of paint when two will do
• Manually performing a process that could be
automated
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Prioritization Matrix
Prioritization Matrix A Prioritization Matrix is typically an L-shaped matrix which makes pairwise comparisons of established
criteria and options. The prioritization matrix is a rigorous method and requires skill to use effectively. It is also
applicable to many situations and has several different configurations. To be used effectively, the criteria and
the options must be clearly developed and a scheme for weighing them must be applied.
Example: A firm is in the process of evaluating three different suppliers for a new product. First, the firm
establishes the criteria that are most important to them and then they rate each supplier against those criteria
on a scale of 1 to 5 (5 being best). The prioritization matrix is given below.
Options/Ratings
Criteria Supplier A Supplier B Supplier C
Competitive costs 4 5 4
Technology growth 3 2 3
Commitment to delivery dates 4 5 5
Quality system 4 4 4
Contributed to future development 3 3 4
Keep in mind that prioritization matrices are typically used in an iterative fashion. For example, the firm will
most likely take the matrix a step further by assigning a weight to each criterion and using that to generate a
final value for each supplier (rating x weight). Then they can sum the values to see which supplier has the
highest priority to be chosen.
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Brainstorming
Brainstorming Brainstorming is a method for generating a large number of creative ideas in a short period of time.
Brainstorming is full of energy, moves rapidly, and is synergistic, creating a large list of ideas which may
eventually be boiled down, or funneled down, to a smaller list of priority items later in the project.
When to use: • When a broad range of options are desired
• When creative ideas are desired
• When participation of an entire team is desired
Procedure: Materials needed: Flipchart (or large dry erase board), pens, tape, and a blank wall (to hang the completed
flipchart sheets).
1. Review the rules for brainstorming with the entire group:
• No criticism. This stifles participation
• No evaluation. Evaluation can be done later
• No discussion of ideas. This tends to slow down the progress of the storm. The idea is to keep
things moving! It’s okay to get clarification, but discussions bog things down
• There are no stupid ideas. The crazier the better. Even if someone suggests an idea jokingly, add
it to the list. All ideas are recorded
• Combining (also known as “piggy-backing”) and expanding on others’ ideas is encouraged.
2. The first thing to do is to review the topic or problem to be discussed. Often it is best phrased as a why,
how, or what question. Make sure everyone understands the subject of the brainstorm. For example:
• Why does the approval process take so long?
• Why is the reader rework so high?
• How can we increase sales?
• What is causing defects in process XYZ?
3. Allow a minute or two of silence for everyone to think about the question.
4. When brainstorming, it is best to solicit the ideas out loud instead of using Post-it notes, or some other
way of anonymously soliciting ideas. If the brainstorming session is done out loud, team members are
able to piggyback on others’ ideas.
5. Invite people to call out their ideas. Quickly. Keep moving. Snap. Snap. One idea on the flipchart after
another. Record all ideas, in words as close as possible to those used by the contributor. Remember, no
discussion or evaluation of any ideas is permitted.
6. Continue to generate and record ideas until several minutes of silence produces no more ideas.
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Brainstorming
Variations: There are many variations of brainstorming. The one described above is sometimes called free-
form, freewheeling, or unstructured brainstorming. Another popular variation is round-robin brainstorming.
In round robin brainstorming, when you invite people to call out their ideas (step 4 above), you have each
person in the group say one idea in turn. If they have no ideas on their turn, they may pass. Stop the round-
robin brainstorming when everyone passes.
Paring down the ideas: Upon completion of the brainstorming session, the team may want to boil down the large list into a smaller,
manageable list. One might ask, if you’re trying to get to a small list, why would you ever brainstorm in
the first place? If the team brainstorms 100 ideas, the probability that the best ideas are on the flipchart
SOMEWHERE is high. If, the brainstorming session was skipped, the probability that the short list will be
missing a key item will be greater than if the team had brainstormed the larger list first.
One quick way of paring down a large list is to give each team member five sticky dots. You tell the team
members that each of them can put their sticky dots on five of the items on the list that they think are the
most important. If a team member feels that one particular item is extremely important, that member might
choose to put all of their five dots on that one item. There is a problem with this method though. It is quick
and therefore minimizes the follow-on (after the brainstorming session) healthy discussions that could make
the paring down process more effective.
Brainstorming with a Cause and Effect Diagram: A good tool to partner with when brainstorming is a cause and effect diagram (sometimes referred to as
a Fishbone Diagram). There is a separate document on the cause and effect diagram supplied with your
materials.
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Cause and Effect Diagram
Cause and Effect Diagram A Cause and Effect Diagram is a graphical tool for displaying a list of causes associated with a specific effect.
It is also known as a fishbone diagram or an Ishikawa diagram (created by Dr. Kaoru Ishikawa, an influential
quality management innovator). The graph organizes a list of potential causes into categories.
The cause and effect diagram shown here happens to have six branches. There is nothing magic about the
number of branches. This one happens to be six because sometimes it’s convenient to have the chart coincide
with the Six M’s. [i.e., Man (in the Generic Sense), Machine, Material, Method, Measurement, and Mother
Nature] The team often chooses to use homogeneous groupings for the branches; however, many groupings
there are, that’s how many branches will be on the chart. (See below)
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Cause and Effect Diagram
Example: A company has experienced problems with a particular machine. The drilling operation is producing
an excessively large size of burr. What’s a burr? If you were to drill a hole in metal; then, if you were to run
your finger across the hole and it cuts your finger, the cut was caused by a burr. It’s virtually impossible to drill
a hole without a burr, but it isn’t impossible to minimize the size of them. The team chose to use a cause and
effect Diagram to list as many causes as possible that might have an effect – which in this case is the burr.
(See below)
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Cause and Effect Diagram
The following illustration is a close-up look at a PORTION of the cause and effect Diagram. As you can see,
the branches are subdivided into smaller branches and twigs. If the original cause and effect Diagram had
more detail, these charts probably would have been called tree diagrams. More detail is always better. And,
that’s why we brainstorm.
Keep in mind that the items listed on the cause and effect Diagram are potential causes. The cause-effect
diagrams should be used not only to document the list of causes, but also to direct data collection and
analysis.
In practice, the team would want to exhaust each of the items listed as potential causes through the use of
the ‘five whys’ technique.
WHY are the instructions inadequate? Because there are no illustrations.
WHY are there no illustrations? There is no reason why we cannot provide illustrations.
WHY not provide illustrations then? We will – starting next month. …and so on. The WHY QUESTIONS
should be reflected on the chart as smaller branches and twigs.
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Affinity Diagram
Affinity Diagram An Affinity Diagram is an analytical tool used to organize many ideas into subgroups with common themes or
common relationships. The method is reported to have been developed by Jiro Kawakita and so is sometimes
referred to as the K-J method.
Example 1: Several members of a small company have just returned from a workshop on the methods of Six
Sigma. On the trip back from the seminar, the group engaged in a vigorous discussion of the challenges they
would confront if they attempted to implement the Six Sigma approach. One person quickly jotted down the
list of challenges they generated. The list of brainstormed challenges is given below.
• Cost accounting discourages other measures • Performance measures discourage cooperation
• Culture does not encourage quality at the source • Poor cooperation among departments
• Distrust of “new initiatives” • Poor opinion of team-based projects
• Inadequate performance reporting tools • Reward systems do not accommodate teams
• No current process champions • Supervisors resistant to required time to train
• Operators not well trained in quality • Suppliers not held accountable for quality
An affinity diagram organizes this list based upon common themes or relationships. For example, an affinity
diagram for this example might look as follows.
Management Training
Poor cooperation among departments
Performance measures discourage cooperation
Poor opinion of team-based projects
Inadequate performance reporting tools
Operators not well trained in quality
Supervisors resistant to required time to train
Culture does not encourage quality at the source
Distrust of “new initiatives”
No current process champions
Systems
Cost accounting discourages other measures
Reward systems do not accommodate teams
Suppliers not held accountable for quality
By organizing the ideas into “affinity groups,” it is much easier to visualize the commonality and plan for and
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Affinity Diagram
Example 2:
Step 1: First, write down the problem. Then quietly put ideas, data, etc. on cards, pieces of paper, or Post-it
notes. The operative word is quietly. This is not like a typical brainstorming session where people are very
vocal about their ideas. We want this to be a quiet exercise so that no one person(s) biases the other team
member’s ideas.
Step 2: Quietly put into homogeneous groupings.
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Affinity Diagram
Step 3: Affinity Heading
Develop affinity heading cards. For example, there is a homogeneous grouping for human resources related
items. There is another grouping for the training department. Another grouping deals with general processing.
One grouping has to do with billing. And, the last grouping addresses employee empowerment. The heading
cards will be placed on top of each of the homogeneous groupings.
Step 4: Put the groupings into the order of the process. For instance, when employees get hired, they first
start off with human resources. The human resources department deals with employee empowerment. And
you have the process itself – that goes in the middle. Billing usually comes late in the game. And finally,
training is something that involves all employees on an ongoing basis so the team chose to put it in last
position.
So what? Now that the team has finished the Affinity Diagram, it is easy to visualize the homogeneous
groupings and therefore might help to guide the team towards a viable project.
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Tree Diagram
Tree Diagram A Tree Diagram is a chart that begins with one central item and then branches into more and keeps branching
until the line of inquiry begun with the central item is exhausted. The tree diagram, with its branching steps,
motivates you to move from the general to the specific in a systematic way.
Example: A company has been losing key employees to competitor firms. It decides to form a task force to
investigate the issue of employee retention and the group prepares the following tree diagram.
Note that the tree diagram deploys the central issue from left to right and moves from a general statement of
the issue to more and more specific lines of thought related to the central issue. A tree diagram is a good tool
to use to organize a team’s thinking about an issue so that the main ideas and relationships are immediately
apparent.
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Interrelationship Diagram
Interrelationship Diagram An Interrelationship Diagram shows graphically the cause-and-effect relationships that exist among a group
of items, issues, problems, or opportunities. It is particularly useful in helping to identify the potential causal
relationships that might lie behind a problem that continues to recur despite attempts to resolve it.
Example: A local physicians’ group is experiencing a relatively high number of patient complaints regarding
the lack of returned phone calls following a patient visit where some kind of test was ordered. In particular,
the patients are frustrated that the promised call notifying them of the test results is either delayed or must
be initiated by the patient. The office manager of the group conducts a brainstorming session to generate
potential reasons for the lack of effective and timely follow-up calls. The group then takes the brainstormed
list and organizes the potential reasons using an interrelationship diagram.
The basic idea is to count the number of “in” and “out” arrows to and from a particular issue and to use
these counts to assist you in prioritizing the issues. In the interrelationship diagram above, “Overly optimistic
promise dates for follow-up calls” is a key issue and, of course, would cause patients to expect a phone call
faster than the group believes it can deliver it. However, do not summarily ignore or devalue the importance of
issues with few “in” and “out” arrows until you have verified empirically the influence of these issues.
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Process Decision Program Charts (PDPC)
Process Decision Program Charts (PDPC) A Process Decision Program Chart (PDPC) is a tool for identifying and documenting the steps required
to complete a process. It is also useful for anticipating any issues and problems that might surface in the
implementation of the process, therefore affording the opportunity to devise countermeasures.
Example: Below is a PDPC for the implementation of statistical process control in the machining area of a
plant.
Note that the process statement is at the top level of the chart and the required process steps move from
left to right at the next level. Further levels of detail may be required until the final level is reached which
indicates countermeasures that might be taken to prevent any delay with the process implementation.
The PDPC chart can be a very effective tool for assuring the process is deployed without costly delays and
problems. It can also be used in the problem solving process where it allows for the identification of potential
process issues that have led to a process failure.
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Activity Network Diagram
Activity Network Diagram An Activity Network Diagram is a diagram of project activities that shows the sequential relationships of
activities using arrows and nodes. An activity network diagram tool is used extensively in project management
and is necessary for the identification of a project’s critical path (which is used to determine the expected
completion time of the project).
Example: Suppose the team is tasked with improving the process of building a house. The team lists the
major steps involved – everything from the excavation step through the landscaping step.
A. Excavate D. Electrical G. Interior
B. Foundation E. Roof H. Exterior
C. Frame F. Masonry I. Landscape
The team creates a chart – Activity Network Diagram – where the nodes (the boxes) represent the nine major
steps involved in building a house. Arrows that connect the nodes show the flow of the process.
Some of the process steps (nodes A, B, and C) run in series, while other process steps (nodes D, E, and F)
run in parallel. Notice that step B cannot happen until step A has been completed. Likewise, step C cannot
happen until step B has completed. Steps G and H cannot happen until steps D, E, and F have completed –
and ALL need to be completed before step H. So, nodes A, B, and C are running in series. Nodes D, E, and F
run in parallel. This is important to know because those steps that are running in parallel most likely will have
different expected completion times.
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Activity Network Diagram
A. Excavate 5 days
B. Foundation 2 days
C. Frame 12 days
D. Electrical 9 days
E. Roof 5 days
F. Masonry 8 days
G. Interior 10 days
H. Exterior 7 days
I. Landscape 5 days
Critical Path The team’s job is to take note of which of the nodes D, E, and F, will be taking the most amount of time,
and which of those nodes is expected to take the least amount of time. This is essential when creating the
Critical Path. For instance, if node D is expected to take the most amount of time as compared with nodes
E and F, it is not important that nodes D and E start at the exact same time as node F. Those steps can
start later, but they have to be finished no later than the most time consuming of the three steps that run in
parallel. Likewise, the team will have to evaluate parallel steps G and H in the same way. The team will need
to understand which step will take the longest to complete and which will take the least amount of time to
complete. Steps G and H do not have to start at the same time, but they have to be finished no later than the
most time consuming of the two steps that run in parallel. The team evaluates the nine steps and come to a
consensus on how many days each of the nine steps will take. The critical path is a line that goes through all
of the nodes that have the longest expected completion times.
Most Likely Time Nodes A, B, and C run in series, so the critical path is straightforward. Notice that between the three nodes
that run in parallel, (nodes D, E, and F) node D is expected to take the longest to complete as compared to
the other two nodes. Notice also that step G takes the longest to complete as compared to step H. The critical
path would run through nodes D and G because those particular nodes have the longest expected completion
times. The line above shows the critical path. By looking at the Activity Network Diagram, the team can easily
see that the expected completion time as defined by the critical path is 43 days (5+2+12+9+10+5 = 43
days). That’s the most likely time.
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Activity Network Diagram
Optimistic Time The team might want to know what the best case (Optimistic Time), in terms of time, would be. To come up
with that number, the team would decide upon the shortest possible time for each of the nodes, and then add
those up. The numbers in parenthesis are the most optimistic times (4+2+10+8+8+4 = 36).
Pessimistic Time The team also might want to know what the worst case (Pessimistic Time), in terms of time, would be. To
come up with that number, the team would decide upon the longest possible time for each of the nodes, and
then add those up. Note: To determine the best case or the worst case, the critical path line must be followed.
The numbers in parentheses are the most pessimistic times (7+3+14+10+11+6 = 51). Remember, you
are only calculating the numbers along the critical path when calculating the most optimistic and pessimistic
times.
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Activity Network Diagram
Expected Time So what does all of this mean? It means the project most likely will take 43 days, but it could take 51 days,
or it can be done as soon as 36 days.
Expected Time = Optimistic + [4(Most Likely)] + Pessimistic
= 6
Expected Time = 36 + 172 + 51
= 43.2 days 6
Control Bands We could calculate control bands around the average. Here’s how we do that:
Limits of expected variation = Optimistic - Pessimistic
= 6
Limits of expected variation = 51 - 36
= 6
Limits of expected variation = 15
= 2.5 6
For the critical path, we can expect the project to take from 40.7 days to 45.7 days
43.2 + 2.5 = 45.7 days on the high side
43.2 – 2.5 = 40.7 days on the low side.
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Project Charter: Description
Project Charter: Description A Project Charter is a document that contains the basic elements (i.e., business case, problem statement,
and scope) of the improvement project and answers the following questions:
1. The Business Case describes what the project does, how it impacts the strategic business
objectives of the company, is used as a motivational tool that describes why the project is worth
doing, and it explains the consequences of not doing the project.
2. The Problem Statement is specific and measurable (quantifiable). It is an indication of how
long the problem has existed, describes the impact to the organization, and describes the gap
between the current state and the desired state.
3. What is the Scope of this project? Other functions and areas of the firm need to know if their
group is likely to be impacted by the improvement project.
In addition to what is listed above, the project charter should serve as a basic working document for the
working team and a source of basic information for the employees of the firm regarding items such as the
following:
• A brief description of the main reasons for pursuing the project and the anticipated benefits to be gained if
the project is successfully completed.
• A list of the milestones of the project and the key deliverables, along with the relative importance of these
deliverables (that is, how much improvement will be accomplished relative to the current state).
• A brief review of the main stakeholders of the project and any issues that are likely to surface that might
require stakeholder participation to resolve.
• A brief listing of any risks to the completion of the project.
It is helpful to remember that the project charter functions as a
“guiding hand” for completion of the project and, as such, incorporates
the basic improvement discipline utilized in all of modern quality: the
Plan-Do-Study-Act cycle. This is sometimes known as the Deming
Wheel or the Shewhart Wheel.
The idea is to use the project charter itself as a guide to managing the
project, meeting deliverables, etc.
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Project Charter: Blank Worksheet
Project Charter: Blank Worksheet The outlines of the typical project charter look like the following:
Project Title: __________________________
Business Case:
Problem Statement:
Project Scope Includes:
Project Scope Does Not Include:
Project Goals:
(1)
(2)
(3)
Anticipated Support or Resources Required:
Project Schedule
Team member
Team member
Team member
Team member
Team member
Team member
Milestones Target Actual Status
(1)
(2)
(3)
Anticipated Financial Impact:
Prepared by: Date:
Approved by: Date:
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Project Charter: Example
Project Charter: Example A manufacturing firm is concerned about increased competition in one of its key markets. For many years
it has realized solid sales and good profit margins on a packaging product for the medical device industry.
Recently, however, the firm has experienced growing competition for this product which uses a specialized
adhesive to bond together packaging films that can then be sterilized. The firm is concerned about both the
quality of the packaging produced (Is it keeping up with competition?) and the cost of the adhesive being
utilized (Is the equipment applying to much?). The operations manager forms a project team to address these
issues. The first business of the project team is to develop a basic project charter.
The firm’s quality and cost reduction project charter was as follows:
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Project Charter: Example
Project Title: Adhesive Improvement
Business Case: The adhesive lamination packaging products are experiencing increasing challenges to
quality and profitability.
Problem Statement: To reduce the cost of production by 2.5% and to improve the capability of key
lamination processes to a Cpk of 1.3 or better.
Project Scope Includes: The current adhesive lamination processes in Plant X manufacturing medical
device packaging.
Project Scope Does Not Include: Nonmedical device products.
Project goals: 7% of total
Cpk = 0.8
Cpk = 1.1
162 hrs/month
4% of total
Cpk ≥ 1.3
Cpk ≥ 1.3
75 hrs/month
(1) Reduce the adhesive costs.
(2) Improve key process capabilities:
Process 1
Process 2
(3) Reduce rework costs.
Project Deliverables:
• Creation of baseline measures of cost and quality
• Assessment of quality (Cpk, etc.) of major competitors
• Update of adhesive lamination costs of quality
Anticipated Support or Resources Required:
Cost accounting personnel
Project Deliverables:
(1) Define: Full project definition.
(2) Measure: Measurement of current state.
(3) Analyze: Complete cost and quality analysis.
(4) Improve: Implementation of basic improvements.
(5) Control: Revised control processes in place.
1/30/20xx
3/15/20xx
4/15/20xx
7/15/20xx
8/21/20xx
Tim Maze, Quality Manager
Phyllis Trask, Operations Manager
John Blum, Operator
Nicole Peterson, Inspector
Kaleb Lile, Finance
Jacob Neil, Engineering
Milestones Target Actual Status
(1)
(2)
(3)
Anticipated Financial Impact:
$2.4 million cost improvement per quarter
Prepared by: Date:
Prepared by: Date:
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Project Charter Highlights
Project Charter Highlights Business Case (nonspecific and nonquantifiable)
• Describes what the project does
• How it impacts the strategic business objectives (in nonquantifiable terms)
• Explains why the project is worth doing
• Explains the consequences of not doing the project
Problem Statement (specific and quantifiable) • Specific
• Measureable
• Describes the impact to the organization (in quantifiable terms)
• Describes the gap between the current state and the desired state
• No causation suggested
• No solution suggested
• No blame given
Questions for the champion to ask: • Impact on the strategic business objectives of the company (scale 1-5)
• Current performance level of this process (scale 1-5)
• Impact of resources (scale 1-5)
• Potential cost savings (scale 1-5)
• Probability of success (scale 1-5)
Note: We need a score >20 (rule of thumb) for a good project.
Example 1:
Business Case: (short and to the point) Not so good: The wait time for customers is too long and it is driving customers away. The team
should not suggest a reason for the problem in the business case, nor in the problem statement.
(There may be some other reason for customers leaving prematurely.)
Better: Customers are leaving before they are served. The project will focus on optimizing the
serving process. (Describes WHAT the project does.)
Best: Customers are leaving before they are served. Not only are we are losing revenue, the word
is spreading that our service is poor. The project will focus on speeding up the serving process.
(Include WHY we are doing this project.)
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Project Charter Highlights
Problem Statement: (in quantifiable terms) What specifically is the problem? Note: In order to write this problem statement data was gathered
from the voice of the customer.
Good : From the time a customer is seated until the menus are delivered and the drink orders are
taken, customers expect to wait no longer than four (4) minutes. Our server process has an average
processing time of seven (7) minutes. (Includes the current state.)
Better: Customers that are seated between 7am and 8:30am expect menus to be delivered and
the drink orders taken in four (4) minutes or less. Our server processing time is averaging seven (7)
minutes during this timeframe. ( Includes the current state.)
Even better: Customers that are seated between 7am and 8:30am expect menus to be delivered
and the drink orders taken in four (4) minutes or less. Our server processing time is averaging
seven (7) minutes during this timeframe. We are losing an average of 24 customers per day.
Average revenue per customer is $16 – annual hard cost loss is $140,160. (Include EXTENT of the
problem.)
BEST: Customers that are seated between 7am and 8:30am expect menus to be delivered and the
drink orders taken in four (4) minutes or less. Our server processing time is averaging seven (7)
minutes during this timeframe. We are losing an average of 24 customers per day. Average revenue
per customer is $16 – annual hard cost loss is $140,160. By cutting the average time from 7
minutes down to 1 minute, we could prevent all of these customers from leaving. (Bridges the gap
between current state and future state and it includes a goal.)
Scope:
What the scope is not:
• It is not merely a timeline (i.e., when the project is to begin and when it is expected to end.)
• It is not restating the problem being attacked.
What the scope could be:
It could be a timeframe (Example: from the time a customer walks through the door to when they
leave the restaurant.) It could even be a subset of that. (Example: all of the steps from the moment
a customer is seated to when they are served the check.)
Not included in this scope:
Quality or quantity of the food. Note: Is quality and quantity of food important? Yes, but that would
be another project at some other time. Let’s say the scope was never defined. The team would start
to work on delivery time. Then they might include the quality of the food, food quantity consistency,
customer comfort, temperature in drinks, room temperature, restroom cleanliness, the sign out front
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Project Charter Highlights
needs to be painted, and on and on. The project would never get done. Plus, how can the team ever
take “the before picture” or develop the process map without first defining the scope?
The scope could also be:
It could be a subset of process steps (Example: from, and including the seating step through, and
including everything up to, but not including the payment of the check. What the scope is intended
to do is place limits on what the team is working on.
Boundaries:
The scope defines the boundaries of the project – usually some beginning point and ending point.
For example, if I were leading a project to improve the delivery of course materials to students, we
might have the following boundaries:
The process (in this project) starts: When the customers say, “Yes, I am interested in enrolling.” The
process (in this project) ends: When the customer receives the box from UPS that contains their
book, CDs, and text. The project stays within that confinement.
Included might be: Enrollment process, warehousing process, accounting process, and UPS delivery
process.
What would NOT be included:
• Errors in the books or CDs
• User friendliness of the materials
• The length of time involved by the enrollment representative to “close the sale” as that would be
just outside of the predefined scope
• The length of time involved by the student to login to the class as that would be just outside of
the predefined scope
• Any of these items would be fine for some other project
Scope creep:
“Scope creep” is a common phenomenon with teams, but should be avoided. If the scope was
predetermined to be as stated above, the team should stick to the confines of that scope. If they do
not stick to it, eventually the team might begin to work on “login times.” Then, they might add to
that the time it takes the enrollment rep to “close the sale.” Then, they might add to that making
sure all materials are delivered error free … and on and on it goes. The scope creeps on and on. The
team soon becomes frustrated and the probably of failure increases. It is best to establish the scope
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Project Charter Highlights
Tollgate #2 of DEFINE – Identify customers’ needs and requirements 1. Who are your customers? (primary and secondary)
2. What is their need? (Usually one word in simple terms)
3. What are their most critical requirements? (In specific terms. What will make your customers satisfied, or
dissatisfied in terms of meeting that need? We will use the Critical to Quality (CTQ) Tree for this.
Tollgate #3 of DEFINE—Process Map (at a high level) 1. Did you name the process? (impacts scope)
2. Did you identify the starting and stopping points of the process? (impacts scope)
3. What are the outputs?
– Brainstorm all outputs. Some will be critical to the customer satisfaction; others will not be so critical.
We will use the critical-to-quality tree to boil these done to two or three later when we make up our
Data Collection Plan.
– Do this step first because everything – suppliers, inputs, and process – leads to the outputs.
4. List out the recipient(s) of the output. Is this customer(s) the same as you had identified in Tollgate #2?
5. Brainstorm a list of the suppliers that provide the inputs to the process.
6. Brainstorm a list of all of the inputs. (whether or not they are deemed critical)
7. List out the 5–7 high level steps in the process (AS IT IS today)
Baseline Project SigmaTollgate # 1 of MEASURE – Data Collection Plan
Meas Type Meas
Type Data
Oper Def
Spec Trgt Coll Form
Samp Base Sigma
1. Measure: Taken from the SIPOC
2. Type of measure: Input, Output, or Process
a. 2-3 Output measures
b. 1-2 Input measures
c. 1 process measure
3. Type of data: Continuous or attribute
4. Operational definition
5. Specification (your customer’s limits of acceptability)
6. Target: Larger is better, Smaller is better, Nominal is best
And, provide an actual value for each target.
7. Data Collection Form (if continuous = histogram or run chart)
If attribute, use Pareto chart or pie chart
8. Sampling (needs to be representative, random, and how much)
9. Baseline Sigma (this is the next tollgate)
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Project Charter Highlights
Tollgate # 2 of MEASURE – Determine Baseline Project Sigma • What is the unit? (i.e., the output of your process)
• Is your customer only interested if the product/service is 100% correct, or are they willing to give you
partial credit if there are one or more defects in an otherwise perfect product or service?
– If they are only interested if the product/service is 100% correct, we will use the DPMU calculation
(defects per million units)
– If they are willing to give partial credit when defects are found, we will use the DPMO calculation
(defects per million opportunities)
•
•
Sigma level without 1.5 σ shift
DPMO without 1.5 σ shift
Sigma level with 1.5 σ shift
DPMO with 1.5 σ shift
1.0 317,311 1.0 697,672
1.5 133,614 1.5 501,350
2.0 45,500 2.0 308,770
2.5 12,419 2.5 158,687
3.0 2,700 3.0 66,811
3.5 465.35 3.5 22,750
4.0 63.37 4.0 6,210
4.5 6.80 4.5 1,350
5.0 0.574 5.0 232.67
5.5 0.038 5.5 31.69
6.0 0.002 6.0 3.40
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Critical-to-Quality Tree
Critical-to-Quality Tree A Critical-to-Quality Tree is a tool used to translate the needs of the customer into measureable product and
process quality characteristics. To generate a Critical-to-Quality Tree, we begin by listing the customer’s needs,
preferably in the customer’s own words. We proceed to refine those needs by asking pertinent questions about
them until we have arrived at quality characteristics that can be measured (product and/or process).
Example: A local bank is interested in improving the training of its tellers. The branch manager decides to first
develop good surveys of customer needs (and customer expectations). To capture this process, the branch
manager uses a Critical-to-Quality Tree as follows.
Note that the arrows indicate that the process of developing a Critical-to-Quality Tree moves from the general
to the specific until you have a list of CTQ characteristics that can be operationally defined and measured.
One of the many benefits of the CTQ process is that it should result in quality characteristics that are readily
transferred to your quality control plan.
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Flow Chart
Flow Chart A Flow Chart (or process flow diagram) is a graphical tool that depicts the distinct steps of a process in
sequential order (from top to bottom of the page). The basic idea is to include any and all steps of critical
importance to the integrity of the process. Also, flow charts are often annotated with performance information.
There are numerous symbols associated with flowcharting and they will differ depending upon the various
industries, but these are the four most common symbols in flowcharting.
Basic Flow Chart Symbols
The start/end symbol obviously begins and ends the flow chart. The process symbol is for listing the steps in
the process. A connector symbol is many times used when the flowchart is too large for one sheet. This circle
connector links where the chart left off on one page to where it picks up again on the next page. The decision
diamond symbol is used where a decision point in the process occurs and it needs to be shown on the chart.
The flow of the chart should go from the top of the page down to the bottom of the page. All decision symbols
should be designed so that the flow reflects the desired state – from top to bottom. Any of the decisions that
do not flow in the desired state, go off to the left side or the right side (instead of going downward) to show
exception handling.
Example: The flow of making coffee is used in this example. The flowchart starts with discarding the coffee
grounds that were left over from the last batch. So far it looks like this. Notice that there isn’t room for the
whole chart on one sheet, so we’re going to use a connector circle to link it to the next page.
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Flow Chart
The connector circle,”A” links the first page with the second
page. So far there has not been a need for any decisions. The
process started, the grounds were discarded, the pot was
rinsed, it was refilled with water, the filter was installed, and
the coffee was put into the filter.
Now we have come down to the first decision diamond. Is the
red light on? Notice that the desired flow is from top to bottom.
If the coffee maker is working, the red light would be lit. The
nondesirable state would be that the red light is not on. Notice
that instead of it flowing in a downward direction, it flows off to
the right. It could have gone off to the left just as easily, but the
team chose for that decision to go off to the right side with a “ no”
response.
Let’s go up to the right side (the
“no” response) and notice that it
goes right into another decision
diamond where it asks if the
coffee maker is plugged in. Notice
again that the desired state faces
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Flow Chart
If the decision diamond that asks whether or not the coffee machine is plugged in comes out the “no” side,
notice that it says to plug the coffee maker in. If the light is on, hit the start button. If the light is still not
working, buy a new coffee maker. Here’s the whole process from Start to End.
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Swim Lane Flow Chart
Swim Lane Flow Chart A flow chart is basically a picture or graph of the sequential process steps of a particular process flow. It is
sometimes called a Process Flow Diagram. The purpose of a flow chart is to clearly document a process by
identifying the essential steps and the relationships ofthosesteps (or subprocesses). It is a very useful tool
for documenting a process in order to train people in the process steps. It is also absolutely essential for a
process that has been chosen for improvement activities.
Below is a partial example of a flow chart that is sometimes referred to as a “swim lane flow chart” utilized
to describe a conference or convention.You can see that the flow chart is constructed in “lanes” that indicate
different responsibilities and processes.
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Swim Lane Flow Chart
When initially developing a flow chart, do not get too preoccupied with obtaining the “correct” end product.
Flow charts tend to evolve as the team considers how the process actually works and the level of detail
required for the chart to be useful for the purposes at hand. It may take several iterations before the team is
satisfied that the flow chart contains the proper level of detail.
One of the great advantages of flow charts is their flexibility. Flow charts essentially begin the analysis of
a process. The swim lane flow chart above might be used by a process analysis team to anticipate points
in the conference where service could fail. For example, a person attending the conference could arrive at
registration and the check-in personnel might not have his or her registration information. This is a service
failure and it could be indicated at the registration point on the flow chart. What caused the failure? The flow
chart can be used to begin the problem solving process. Maybe the personnel at check-in overlooked the
material? Maybe there was a problem with the computerized registration? Whatever the case, the flow chart is
essential to begin the problem solving and corrective action.
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Basic Statistics Using the Texas Instruments TI-30 Calculator
Basic Statistics Using the Texas Instruments TI-30 Calculator (~$15 at your favorite electronics store, or department)
The version on the left has been around forever—probably 20 years. The newer version (right) is quite
different, but much easier to use. This document will show how to compute some basic statistics (i.e., the
mean, standard deviation for a sample, and for a population) for this set of data using a Texas Instruments
30X calculator (both versions):
5 2 4 3 3 6
Note: The only reason the particular model (Texas Instruments TI-30) was chosen is because it is year-after-
year the most commonly found inexpensive scientific calculator.
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Basic Statistics Using the Texas Instruments TI-30 Calculator
Instructions for the older version
To enter the data: 5 2 4 3 3 6
Enter each data point by typing them in one at a time and hitting the [∑+] button after each one. NOTE: The
display will show you the number of values in the statistics register after you enter each one: n = 1, n = 2,
and so on until you see n = 6. Follow the steps, starting with #1. After completing certain steps, (e.g., 1,
3, 5, etc.) the display should appear as illustrated to the right-hand side of the illustration. Note: Depending
upon the initial state of your calculator, you may get an error message after #3a. That’s expected. Just
proceed to #3b, and follow the numbers.
To find the average for the data entered: The function key for the average is in yellow above the [x2] key. To activate it, hit the yellow [2nd] button one
time and then hit the [x2] key. In this example the average is 3.83333333. Refer to the completion of step
#19 in the illustration.
To find the standard deviation for a set of sample data: If the data are from a sample, you will use the [σxn-1] function above the [] key. Since the [σxn-1] function is
in yellow, you will need to hit the yellow [2nd] button one time and then hit the [ ] key. In this example the
sample standard deviation is 1.471960144. Refer to the completion of step #21 in the illustration.
To find the standard deviation for a set of population data: If the data are from a population, you will use the [σxn] function above the [÷] key. Since the [σxn] function is
in yellow, you will need to hit the yellow [2nd] button one time and then hit the [÷] key. In this example the
population standard deviation is 1.34370962472. Refer to the completion of step #21 in the illustration
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Basic Statistics Using the Texas Instruments TI-30 Calculator
To enter the data: 5 2 4 3 3 6
(follow the numbers 1 – 23)
To clear the statistics register: There are two ways:
• Turn the calculator off using the [OFF] key. (Upper right-hand corner of the calculator)
• Use the “Clear Statistics Register” [CSR] function located in yellow above the [7] key. Hit the yellow [2nd]
button one time and then hit the [7] key. Refer to the completion of step #19 in the illustration. Refer to
the completion of step #3a in the illustration. Caution: Always be sure to clear the statistics register.
If you forget, you will notice that upon entering the first number, the display window (should = 1) will
be a continuation of the last time the calculator was used. In this case, the first number entered would
result in a “7” in the window—instead of “1”.
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Basic Statistics Using the Texas Instruments TI-30 Calculator
Instructions for the newer version
To enter the data: 5 2 4 3 3 6
This version of the TI-30 is quite different than the older version. You start by turning the calculator ON. You
do so by pressing the ON button in the lower left-hand corner. Now you need to turn on this STAT function. To
do that, press the 2nd button in the upper left-hand corner; then press the key labeled DATA. In the window
of a calculator, there are two options. The first option is if you want to enter data for ONE variable. The other
choice is if you were going to enter data where there are TWO variables – such as a paired comparison.
Leave it in the default selection (one variable); hit the ENTER key located in the lower right-hand corner
of the calculator. We are going to enter some data, so that is the next button to push – the DATA button
located in the upper middle of the calculator. The cursor in the window will be blinking for you to enter the
1st value – which is a 5. Hit the 5 button. Press on the down arrow on the calculator (located in the upper
right-hand corner. It will say frequency equals 1 in the window. There is only one “5” in the data set, so that
would appear as a “1”. Hit the down arrow again and in the window of the calculator is prompting you to
enter your second data value which is a “2”. Again, hit the down arrow. It will be prompting for the frequency.
There is only one “2” in our data set, so we will leave that as a 1. Hit the down arrow. The next value is a 4.
Press the 4 key. Press the down arrow. Frequency is 1. Hit the down arrow again. Notice that there are two
3’s in the data set. Press the 3 button. Press the down arrow. This time, since we have two 3’s, make the
frequency equal to “2”. Hit the down arrow. It is asking for the 5th value. But, there are six values. Confused?
Remember, we entered the number 3 with a frequency of 2, so we are actually adding our 5th unique value.
Press the 6 button and then press the ENTER key.
All of the data has now been entered, and now you can find some basic statistics for this data set. To do
that, press the STATVAR button located in the upper middle section of the calculator. It starts in the window
by showing you what “n” is equal to. At the lower part of the window, there is a “6” showing. It is showing
you that you have entered 6 values. Press the right arrow button once. Now you are looking at the arithmetic
mean which is 3.83333. Press the right arrow again. You will see the standard deviation of the SAMPLE
which is 1.47196. This is not to be confused with SIGMA of the data. If that were the case, we would be
considering the POPULATION. If that truly was what you were looking for, press the right arrow again and you
will see sigma of the data which is 1.3437.
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Basic Statistics Using the Texas Instruments TI-30 Calculator
To enter the data: 5 2 4 3 3 6
(follow the numbers 1 – 12)
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Standard Deviation: 6 Steps to Calculation
Standard Deviation: 6 Steps to Calculation The formula for Standard Deviation depends on whether you are analyzing population data, in which case it is
called σ or estimating the population standard deviation from sample data, which is called s:
The steps to calculating the standard deviation are:
1. Calculate the mean of the data set (x-bar or µ)
2. Subtract the mean from each value in the data set
3. Square the differences found in step 2
4. Add up the squared differences found in step 3
5. Divide the total from step 4 by either N (for population data) or (n – 1) for sample data(Note: At this point
you have the variance of the data)
6. Take the square root of the result from step 5 to get the standard deviation
Example:
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Standard Deviation: 6 Steps to Calculation
Step 1: The average depth of this river, x-bar, is found to be 4’.
x Step 2: ( x - )
Step 3: ( x - )2
0.5 4 -3.5 12.25
5 4 1 1
6 4 2 4
10 4 6 36
5 4 1 1
1 4 -3 9
0.5 4 -3.5 12.25
SUM: Step 4: 75.50
Step 5: The sample variance can now be calculated:
Step 6: To find the sample standard deviation, calculate the square root of the variance:
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Standard Deviation: Description and Example
Standard Deviation: Description and Example The Standard Deviation of a set of data describes the amount of variation in the data set by measuring, and
essentially averaging, how much each value in the data set varies from the calculated mean.
The formula for standard deviation depends on whether you are analyzing population data, in which case it is
called σ or estimating the population standard deviation from sample data, which is called s:
Example:
The average depth of this river is 4’.
The standard deviation, s, is found to be:
This means that the “typical” depth sample varies from the average depth by 3.55 feet.
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Histograms
Histograms Histograms are used to show the distribution of a set of collected data.
Generic bar charts can be created for any type of categorical or numerical data. If the categories of the data
plotted in a bar chart have no meaningful order, many different charts can be created by rearranging the
order of the bars. Histograms, on the other hand, are used to show the pattern or the distribution of the data
across the categories, so there must be only one correct pattern. For this to be true, the data charted must
be from ordered categories (or classes). The axis that lists the data categories (typically the horizontal axis)
must then be shown in the logical order of the data in order for a histogram to be meaningful. For this reason,
histograms are typically only used with numerical data, although ordinal categorical data can also be used.
The following histogram displays the frequency distribution for this set of Waiting Times data: Note: The
number of cells that reflect the data has an effect on the shape. There is another topic in this program that
discusses those effects.
CUSTOMER WAITING TIME IN SECONDS (n=32)
10.4 12.0 18.7 15.9 11.8 12.0 17.5 11.3
10.9 12.4 11.4 10.7 10.2 13.9 13.0 12.7
12.5 14.3 10.4 16.4 11.4 10.6 13.9 11.2
17.3 11.4 11.2 20.3 19.9 20.0 14.2 11.6
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Cell Intervals: Impact on Shape
Cell Intervals: Impact on Shape The choice of Cell Intervals (the number and width of the classes used to construct a histogram) can have an
effect on the shape of the histogram. In general, a smaller number of intervals results in a less distinct shape,
while a larger number of intervals can reveal unexpected patterns.
Note: Please refer to the other topic on cell intervals for more information about rules-of-thumb in regard to
cell choices.
Example: A random sample of n = 400 people in a small town were asked to state their most recent annual
income. The researchers decided to display the data with a histogram. The annual incomes vary from
$30,000 to $84,000. Unsatisfied with their initial histogram, they increased the number of cell intervals. See
the results below.
While each of the histograms indicates
the data might be bell-shaped or normally
distributed, as the class size increases,
there is more refinement in the shape of
the data. The middle histogram appears
to indicate the data is quite clearly
normally distributed. However, the last
histogram (which has the most number
of cell intervals) indicates another peak in
the data around $66,000 or $68,000.
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Cell Intervals: Rules of Thumb
Cell Intervals: Rules of Thumb When constructing a histogram (a graph in which the classes are identified on the horizontal axis and the
class frequencies are shown on the vertical axis by the heights of the vertical bars) it is important to choose
an appropriate number of Cell Intervals (the grouping you will use to generate the classes for the histogram).
As a rule of thumb, the number of cell intervals “K” can be chosen to equal the smallest whole number that
makes 2K greater than the total number of measurements (n). However, this is not a hard and fast rule, so you
may want to try a few different options (say, K – 1 intervals or K + 1 intervals) as well. Your interval length
can then be computed to be (largest value – smallest value) divided by K.
Example: Suppose we have a group of data from a test administered to new employees to evaluate their
understanding of a new financial product. The highest score possible on the test is 40. The results from 30
employees show a low score of 17 and a high score of 39.
To find the number of cell intervals to use, we would choose the value of K that makes 2K just greater than
n = 30. Thus K would equal 5 (since 25 = 32). Thus we want 5 cell intervals and the width of our cell
intervals is approximately:
Always “round up” to the next whole number. Notice in the example above that quotient came out to 4.4.
Normally, we would round down to four, but if we did that we would only have four in each cell – which
would be insufficient to cover the whole range of numbers 17 to 39. See below.
17 to 20 would be the first cell.
21 to 24 would be the second cell.
25 to 28 would be the third cell.
29 to 32 would be the fourth cell.
33 to 36 would be the final cell, which would make us short on the last cell because we needed values to
cover through that number 39. So, always round up to the whole number.
Let’s use five cells (intervals) with five values in each cell. It would look like this:
17 to 21
22 to 26
27 to 31
32 to 36
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Cell Intervals: Rules of Thumb
You could have chosen to use eight intervals instead of five, and if you would have chosen eight it would have
looked like this:
17 – 19
20 – 22
23 – 25
26 – 28
29 – 31
32 – 34
35 – 37
38 – 40
Notice that the last range goes to 40 which is one higher than 39, but that’s okay because eventually if we
were to continue to collect data we might eventually see a 40.
So, we gave you two examples – one with five intervals, and the other example with eight intervals in it. So,
how do you know when to choose five, when to choose six, or when to choose seven, eight, or nine? There is
no hard and fast rule on this, but here is a “rule of thumb” list.
Number of measurements (n) Cell choices
If less than 50 Use 5 to 7
If 51 to 99 Use 6 to 10
If 100 to 250 Use 7 to 12
If more than 250 Use 12 to 20
Now we count the number of scores that fall into each cell and we are ready to generate the frequency
distribution or histogram. See below.
X
X X X X
17–19 20-22 23-25 26–28 29–31 32–34 35–37 38-40
17 18 19
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Stem-and-Leaf Plots
Stem-and-Leaf Plots Stem-and-Leaf Plots are a valuable combination of a check sheet and a histogram in which the actual data
values are recorded so that the raw data is maintained and the distribution of the data is also shown. The
data are divided into leading digits (“stems”) and trailing digits (“leaves”). For example, the number “17.4”
could be broken into a stem of “17” and a leaf of “4.” The key to effective stem-and-leaf plots is that the
leaves be evenly aligned across all the stems so that the length of each row provides an accurate picture of
the relative frequency of that stem without having to count the actual number of leaves.
Example: Given the data below:
CUSTOMER WAITING TIME IN SECONDS (n=32)
16.8 17.5 12.8 13.2 13.3 17.5 17.7 13.7
14.0 19.8 15.7 13.9 16.8 18.3 19.9 20.6
18.9 17.4 14.1 17.3 18.0 11.2 14.4. 13.7
12.3 13.2 16.8 12.1 16.8 13.2 17.1 18.1
By looking at the data, can you tell if it is normally distributed? Can you tell if you have common cause or
special cause variation? You could use a histogram, but histograms do not preserve the raw data. Plus, a
histogram is usually constructed after the data has been collected which means that it is difficult to get a
sense of the shape of the distribution until after all the data has been collected. You could use the Frequency
Distribution Check Sheet which is fine if there are no cell intervals. In other words, if every time you place
an X on the frequency distribution check sheet, it represents one number, the raw data would be preserved.
The problem is that Frequency Distribution Check Sheets many times will use an X to represent an interval
(e.g.,16.1 through 16.5). All that would be preserved would be that the original data was somewhere
between 16.1 and 16.5. An alternative is to use a Stem and Leaf Diagram (a.k.a. Stem and Leaf Chart, or
Stem and Leaf Plot).
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Stem-and-Leaf Plots
Here is one alternative for a stem-and-leaf plot:
Stems Leaves Frequency
(Count)
10
11 2 1
12 1 3 8 3
13 2 2 2 3 7 7 7 9 7
14 0 1 4 3
15 7 1
16 8 8 8 8 4
17 1 3 4 5 5 7 6
18 0 1 3 9 4
19 8 9 2
20 6 1
TOTAL: 32
Notice by looking at the stem and leaf chart that there appears to be a bimodal distribution meaning that two
things are affecting the process – each one appearing to be normally distributed in and of itself, but when all
of the data are combined it takes on a bimodal shape. You would not know that by just looking at the raw
data. And the beauty of the Stem and Leaf Chart is that the raw data is preserved, while at the same time
giving you a visualization process variation behavior as the process unfolds.
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Box and Whiskers Plot
Box and Whiskers Plot A Box and Whiskers Plot is a graphical tool of exploratory data analysis that allows the comparison of
groups by constructing a graph around five measures for each group: (1) the median, (2) the maximum, (3)
the minimum, (4) the first quartile or 25th percentile, and (5) the third quartile or 75th percentile. This tool
provides visibility of variation among items being evaluated and makes comparisons quick and easy.
Example: Suppose you are interested in comparing the productivity (as measured in volume generated).
Over the course of a month, you randomly select nine days of production for each of the four machines (and
record the number of casting produced that day) which yields the data below. By looking at the data sets for
the four machines, which machine has the most variation; which machine has the least amount of variation;
which one is producing closest to the target value (which is a larger is better quality characteristic)? Of four
machines, which machine is performing the best overall? It's difficult to tell by just looking at the numbers.
On the next page, you'll see the box and whiskers plot that will make these comparisons easy to see.
Machine 1 Machine 2 Machine 3 Machine 4
404 423 436 454
381 436 369 483
394 393 452 500
461 413 531 442
407 447 354 393
396 369 440 475
432 495 428 480
393 390 435 505
443 441 351 464
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Box and Whiskers Plot
A box and whiskers plot using MINITAB is given below.
The vertical lines run from the minimum to the maximum value. This is useful for finding the extreme values.
The horizontal line indicates the median of each group. This is useful to compare the central tendencies of the
four machines.
The bottom and top of the boxes indicate the 25th and 75th percentiles, respectively. This is useful for
visualizing the bulk of the spread of variation among the four machines.
Since productivity is a larger is better quality characteristic, one can see that machine #4 tends toward larger
values, while Machine #1 appears to have the least variation.
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Check Sheets: Defect Location Check Sheet
Check Sheets: Defect Location Check Sheet A Defect Location Check Sheet (also known as a defect map or a measles chart) is a structured, prepared
form for collecting and analyzing data that provides a visual image of the item being evaluated so that data
can be collected visually rather than with words. Check sheets are generic tools that can be adapted for a
wide variety of purposes. The following is an example of check sheet used to collect data on the location of
defects on a product.
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Run Chart: Basic Construction
Run Chart: Basic Construction A Run Chart is a basic graph that displays data values in a time sequence (the order in which the data were
generated). A Run Chart can be useful for identifying trends or shifts in process.
Example: A supervisor of a customer service center collects data on the number of complaints that are filed
each month. Data for the last several months are shown below.
Complaints 4 3 6 5 5 8 2 4 3
Month Jan Feb Mar Apr May Jun Jul Aug Sep
A Run Chart for this data is given below.
Note that the data appear to “move around” with an average of about four complaints filed per month and
there are no obvious trends (sequential moves up or down) present.
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Run Chart: Shifts
Run Chart: Shifts A Run Chart is a basic graph that displays data values in a time sequence (the order in which the data were
generated). A run chart can be useful for identifying shifts and trends.
Example: A supervisor of a customer service center collects data on the number of complaints that are filed
each month. Data for the last several weeks are shown below.
Complaints 4 3 5 3 4 6 4 8 9 10 8 9 9 10 11 9
Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Run Chart for this data is given below.
The run chart shows that for the first seven weeks, the number of complaints moved up and down around an
average just above four. However, for the last several weeks, the number of complaints has shifted upward
and now moves up and down around an average of about nine. The overall average number of complaints
for the sixteen weeks is about seven. A general rule of thumb is when seven or eight values are in succession
above or below the average of the group, a shift has occurred. This is like flipping a coin and seven times in a
row it comes up heads or tails. Could it happen randomly? Yes, but not likely.
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Run Chart: Trends
Run Chart: Trends A Run Chart is a basic graph that displays data values in a time sequence (the order in which the data were
generated). A Run Chart can be useful for identifying shifts and trends.
Example: A supervisor of a customer service center collects data on the number of complaints that are filed
each month. Data for the last several months are shown below.
Complaints 3 2 3 4 6 7 9 10 11
Month Jan Feb Mar Apr May Jun Jul Aug Sep
A Run Chart for this data is given below.
Note that the Run Chart shows a distinctive trend upward in the data. Clearly, a Run Chart would also be
useful in identifying downward trends in the data as well. While different practitioners use different rules, the
basic rule of thumb is when a Run Chart exhibits seven or eight points successively up or down, then a trend
is clearly present in the data. This is like flipping a coin and seven times in a row it comes up heads or tails.
Could it happen randomly? Yes, but not likely.
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Target Values: Illustration 1
Target Values: Illustration 1 The Target Value of a process is the numerical aim of the process that is preferred for the quality
characteristic of interest. Target Values can be of several types, but the most common ones are: (1) smaller is
better, (2) larger is better, and (3) nominal is best. When you think of nominal is best, think of "named value."
Example: Below are three examples of Target Values, one each for (1) smaller is better, (2) larger is better,
and (3) nominal is best.
Smaller is Better An ambulance service is in the process of analyzing its response times. It has a large
amount of past data and so they decide to use it to set a maximum response time
limit of 9.5 minutes. In this scenario, the smaller the response time the better the
performance of the ambulance service.
Larger is Better A producer of circuit boards has measured the amount of force it takes to break the
circuit board away from the electronic controller. In this scenario, the larger the force
required to break the circuit board away from the controller the better.
Nominal is Best A mortgage lending firm has done extensive testing of its new loan analysis process
and believes that a well-trained mortgage consultant should take between 2.25 hours
and 3.25 hours to analyze a certain class of mortgage loans. Their research shows
that completing the analysis in less than 2.25 hours usually means the analysis is not
thorough enough and taking longer than 3.25 hours means the mortgage consultant
is not efficient enough. Therefore, the closer to an average of 2.75 hours, the more
confident they are that the right balance between thoroughness and efficiency has been
achieved.
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Target Values: Illustration 2
Target Values: Illustration 2 The Target Value of a process is the numerical aim of the process that is preferred for the quality
characteristic of interest. Target Values can be of several types, but the most common ones are: (1) smaller is
better, (2) larger is better, and (3) nominal is best.
Example: Below are three examples of Target Values, one each for (1) smaller is better, (2) larger is better,
and (3) nominal is best.
Smaller is Better A producer of packaging measures its machine centers on the percent of waste
(packaging that cannot be used due to start-up, shut-down, or quality problems) that
is generated for each type of packaging produced. In this scenario, the smaller the
amount of waste generated the better.
Larger is Better A bank regularly tracks customer satisfaction scores received from formal surveys. The
best customer satisfaction score the bank can receive is a 10. In this scenario, the
larger the score (that is, the closer the average score is to ten) the better the bank is
performing.
Nominal is Best A manufacturer has solid product performance and engineering test data that shows a
particular metal shaft they produce must be between 9 and 11 millimeters in diameter
to perform well in the intended application with 10 millimeters being optimal. In this
scenario, the nominal value of 10 millimeters is best.
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Target Values: Illustration 3
Target Values: Illustration 3 The Target Value of a process is the numerical aim of the process that is preferred for the quality
characteristic of interest. Target Values can be of several types, but the most common ones are: (1) smaller is
better, (2) larger is better, and (3) nominal is best.
Example: Below are three examples of Target Values, one each for (1) smaller is better, (2) larger is better,
and (3) nominal is best.
Smaller is Better A company that assembles small gift sets that parents can purchase and have sent to
their children in college tracks the quality of the assemblies by randomly selecting a
fixed number of completed gift boxes and counting the number of non-conformances in
the assembly of the gift box (for example, items placed in the wrong compartments).
The rating is tracked as the number of non-conformances per 100 gift boxes
assembled. In this scenario, the target value set by the company will be smaller is
better.
Larger is Better A supplier of food storage bags submits a random sample of the bags to a pressure
test and records the pressure (in psi) required to burst each bag. Since the higher the
pressure required to burst the bag, the stronger the sealing capacity of the bag, this is a
larger is better target value.
Nominal is Best A company tracks the amount of lubricant required to keep a bearing system in
its machining centers functioning optimally. Too much lubricant and the expensive
chemical is wasted. Too little and the machines wear out the bearings. Since there is
an optimal amount of lubricant that should be administered, this is a nominal is best
target value.
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DPU, DPMO, PPM, and RTY
DPU, DPMO, PPM and RTY Four common measures of process performance are Defects Per Unit (DPU), Defects per Million
Opportunities (DPMO), Parts per Million Defective (PPM), and the Rolled Throughput Yield (RTY). The key
to understanding the difference between these terms is to understand the difference between a “defect” and a
“defective” item:
• A defect refers to a flaw or discrepancy on an item where more than one flaw (defect) can be found. For
example, a hospital admission form contains several fields of information that can be missing or incorrect,
so a given form can have more than one defect. This means that a sample of 10 forms can show more
than 10 defects.
• An item is said to be defective when the decision is made that the item is not acceptable, based either
on one characteristic or the accumulation of multiple defects. This means that a sample of 10 items can
show a maximum 10 defective units.
Defects per unit (DPU) – the average number of defects per unit of product.
For example when 26 defects (flaws) are found on 10 units of product, the DPU is 26/10 or 2.6 defects per
unit.
Defects per Million Opportunities (DPMO) – a ratio of the number of defects (flaws) in 1 million opportunities
when an item can contain more than one defect. To calculate DPMO, you need to know the total number of
defect opportunities.
For example, a form contains 15 fields of information. If 10 forms are sampled and 26 defects are found in
the sample, the DPMO is:
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DPU, DPMO, PPM, and RTY
Parts per Million Defective (PPM) – the number of defective units in one million units. (PPM is typically used
when the number of defective products produced is small so that a more accurate measure of the defective
rate can be obtained than with the percent defective.)
For example, a sample of 50 cell phones finds that 3 of them are defective. The PPM defective is then:
Rolled Throughput Yield (RTY) (also known as the First Pass Yield) – the probability (or percentage of time)
that a manufacturing or service process will complete all required steps without any failures. Reliability
principles are the basis for calculating the rolled throughput yield. The reliability formula for a system in series
with n process steps is:
Rs = (R1) (R2) (R3) (R4) … (Rn)
Since the reliability of a process step is the yield of that process step when quality is the performance metric,
this formula then becomes:
RTY= (Y1) (Y2) (Y3) (Y4) … (Yn) where Y is the yield (proportion good) for each step
For example, a four-step process has a yield of 0.98 in step 1, 0.95 in step 2, 0.90 in step 3, and 0.80 in
step 4.
RTY = (0.98)(0.95)(0.90)(0.80) = 0.67032
This means that only 67.032% of the units completed on this process will make it through all four steps
without needing any rework or repair.
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DPMO to Sigma Level Relationship
DPMO to Sigma Level Relationship Sigma level without
1.5 σ shift DPMO without
1.5 σ shift Sigma level with
1.5 σ shift DPMO with 1.5 σ shift
1.0 317,311 1.0 697,672
1.5 133,614 1.5 501,350
2.0 45,500 2.0 308,770
2.5 12,419 2.5 158,687
3.0 2,700 3.0 66,811
3.5 465.35 3.5 22,750
4.0 63.37 4.0 6,210
4.5 6.80 4.5 1,350
5.0 0.574 5.0 232.67
5.5 0.038 5.5 31.69
6.0 0.002 6.0 3.40
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Pp and Ppk: Formulas
Pp and Ppk: Formulas The Pp and Ppk indices estimate the performance of the process based on the current overall level of
variation. They are considered to be more accurate measures of the actual performance of the process since
they reflect how the process is actually performing rather than how it will perform if capable. For this reason,
the Pp/Ppk indices are referred to as process performance indices. Note that the more stable the process is,
the closer the performance indices (Pp/Ppk) will be to the capability indices (Cp/Cpk).
Because the Pp/Ppk indices measure actual process performance, the standard deviation used is always
calculated from the entire set of process sample data as follows:
Since the standard deviation is a measure of the variation over the entire process, it is known as the overall
variation.
The Pp compares the total predicted process variation (defined as + / - 3 standard deviations) to the
allowable process variation (specification range):
The Ppk compares the actual process center and spread to the nominal or target process center and spread:
USL: Upper Specification limit
LSL: Lower Specification Limit
Ppk is based on the distance from the process mean to the nearest specification limit (USL or LSL), and
therefore riskiest, performance target, so the smallest value is always selected.
Pp and Ppk indices are a "larger is better" target . With Six Sigma, the goal is generally to reach a level of 1.5
and greater. However, the target of an acceptable level of performance really depends on an organization's
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Pp and Ppk: Example
Pp and Ppk: Example An order-entry process is required to be performed on an average of 7 + / - 2 minutes. A control chart was
placed on the process and 25 sample groups of 5 items each were collected. It was found that the process
was stable and normally distributed with an average time of 5.6 minutes and standard deviation of 1.74
minutes.
A Pp of 0.38 means that the tolerance interval is less than half the width of the total process variation (or the
process variation is 2.63 times the size of the tolerance interval). This means that this process will have a
large percentage of orders entered outside of the target time frame even if the average processing time can be
centered in the middle of the target interval. A Ppk of 0.11 means that only about 11% of a normal curve will
fit between the average and the nearest performance limit.
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Finding a Z Score: Example 1
Finding a Z Score: Example 1 Hospital stays for admitted patients at a certain hospital are measured in hours and were found to be
normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these
stays can be expected to last for longer than 300 hours?
SOLUTION: Using the formula for determining the Z statistic yields:
1. Using a Standard Normal Table that shows the total probability less than any given Z value (see the
example for Form 1) yields a probability of 0.9082 or 90.82% below 300 hours. Since the question asks
how many will stay longer than 300 hours, the percent of patients staying longer than 300 hours is found
by subtracting 90.82% from 100%. The final answer is therefore 9.18%.
2. Using a Standard Normal Table that shows the area under the curve between the mean and Z (see the
example for Form 2) yields a probability of 0.4082. This means that 40.82% of the patients can be
predicted to stay between 200 and 300 hours. Since the question asks how many will stay longer than
300 hours, and since the mean of a Normal Curve divides the distribution into two equal halves of 50%
each, the percent of patients staying longer than 300 hours is found by subtracting 40.82% from 50%.
The final answer is therefore 9.18%.
3. Using a Standard Normal Table that shows the total probability greater than any given Z value (see the
example for Form 3) yields a probability of 0.0918 or 9.18% above 300 hours. Since the question asks
how many will stay longer than 300 hours, nothing further needs to be done to this value. The final
answer is therefore 9.18%.
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Finding a Z Score: Example 2
Finding a Z Score: Example 2 Hospital stays, for admitted patients at a certain hospital are measured in hours and were found to be
normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these
stays can be expected to last less than 75 hours?
SOLUTION: Using the formula for determining the Z statistic yields:
1. Using a Standard Normal Table that shows the total probability less than any given Z value (see the
example for Form 1) yields a probability of 0.0475 or 4.75% below 75 hours. Since the question asks
how many will stay less than 75 hours, nothing further needs to be done to this value. The final answer is
therefore 4.75%.
2. Using a Standard Normal Table that shows the area under the curve between the mean and Z (see the
example for Form 2) yields a probability of .4525. (Note that this form of Z table does not show any
negative Z scores. Since the Normal Distribution is symmetric about the mean, values the same distance
below and above the mean are equal.) This means that 45.25% of the patients can be predicted to stay
between 200 and 75 hours. Since the question asks how many will stay less than 75 hours, and since
the mean of a Normal curve divides the distribution into two equal halves of 50% each, the percent
of patients staying less than 75 hours is found by subtracting 45.25% from 50%. The final answer is
therefore 4.75%.
3. Using a Standard Normal Table that shows the total probability greater than any given Z value (see the
example for Form 3) yields a probability of 0.9525 or 95.25% above 75 hours. Since the question asks
how many will stay less than 75 hours, 0.9525 must be subtracted from 1 to obtain the answer of
0.0475. The final answer is therefore 4.75%.
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Finding a Z Score: Example 3
Finding a Z Score: Example 3 The mean inside diameter of a sample of 200 washers produced by a machine is 0.502 inches and the
standard deviation is 0.005 inches. The purpose for which these washers are intended allows a maximum
tolerance in the diameter of 0.496 to 0.508 inches, otherwise the washers are considered defective.
Determine the percentage of defective washers produced by the machine, assuming the diameters are
normally distributed.
SOLUTION: Since we are concerned about those washers whose diameter is less than the specification limit
and those greater than the specification limit we must compute two Z scores:
1. Using a Standard Normal Table that shows the total probability less than any given Z value (see the
example for Form 1) yields a probability of .1151 or 11.51% below .496”, which means that 11.51%
are defective on the low side. The table also shows 0.8849 or 88.49% below 0.508”. Since the question
asks how many will be defective (above 0.508”), we subtract 0.8849 from 1 to get the answer of
0.1151 or 11.51% defective on the high side. Since both sides produced defective parts, these two
values are added together to obtain the final answer of 23.02%.
2. Using a Standard Normal Table that shows the area under the curve between the mean and Z (see the
example for Form 2) yields a probability of 0.3849. (Note that this form of Z table does not show any
negative Z scores. Since the Normal distribution is symmetric about the mean, values the same distance
below and above the mean are equal.) This means that 38.49% of the washers will be between 0.502”
and both 0.496” and 0.508”, for a total of 76.98% within the specification limits. Since the question
asks how many will be defective, the percent of good washers, 76.98% is subtracted from 100% to
obtain the final answer of 23.02%.
3. Using a Standard Normal Table that shows the total probability greater than any given Z value (see the
example for Form 3) yields a probability of 0.1151 or 11.51% above 0.508” and 0.8849 or 88.49%
above 0. .496”. Since the question asks how many will be defective, 0.8849 must be subtracted from 1
to obtain the answer for the percent defective on the low side, or 0.1151 (11.51%). The final answer is
obtained by adding the 11.51% defective on the high side to the 11.51% defective on the low side for a
total of 23.02%.
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Finding a Z Score: Example 3
Using Form 2 of the Normal Table a value of 1.20 on the z-table shows a probability of 0.3849. Since
this particular z-table is based on 50% of the population, we can expect 0.5000 minus 0.3849 proportion
(11.51%) of the washers to be defective on the high side. Since the distribution is symmetrical we can
conclude that 11.51% are also defective on the low side. Thus we have a total percent defective of 11.51%
plus 11.51% or 23.02% defective washers.
Using Form 3 of the Normal Table a value of 1.20 on the z-table shows a probability of 0.8849. This table
is similar to Table 1 except that it only has values of Z greater than 0.0, it has no negative values. Since this
particular z-table is based on probability less than any given z-value and we are interested in the probability
of greater than the z-value we subtract the .8849 from 1.00 to get the percent of washers defective on the
low side (11.51%). Since the Normal distribution is symmetrical, the probability a z-score less than minus
1.20 is also 11.51%. Adding these two probabilities together we can expect 23.02% of the washers to be
defective.
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Z Table: Form 1
Z Table: Form 1 The normalized Z table appears in three basic forms. The three basic forms are:
• Form 1 – Probability from negative infinity to a calculated Z value. Typical tabulated Z values
include -3.59 to +3.59 by increments of 0.01
• Form 2 – Probability from a Z value of 0.00 to a calculated Z value. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01. This document can be found in a topic sheet
for form 2
• Form 3 – Probability from a calculated Z value to positive infinity. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01. This document can be found in a topic sheet
for form 3
Regardless of the form used, a correct answer can be obtained when any of these three forms is properly used
to find normal probabilities. The following is an example of a Form 1 z table which shows the probability
from negative infinity to the calculated Z value. The Z value is calculated to 2 decimal places. The whole
number and first decimal place are found in the first column labeled “z” and the second decimal place is
found in one of the adjacent columns labeled “x.x0,” “x.x1,” x.x2,” etc. The value at the intersection of the
row and column selected is the area under the curve (probability) between negative infinity and that Z score.
Example: A Z score is calculated to be -2.45. The value in this table at the intersection of -2.4 and x.x5 is
0.0071, which means that there is a 0.0071 (or .71%) probability of a value falling between negative infinity
and a Z score of -2.45.
Example: A Z score is calculated to be 0. The value in this table at the intersection of -0.0 and x.x0 is
0.5000, which means that there is a 0.5000 (or 50%) probability of a value falling between negative infinity
and a Z score of 0.
Example: A Z score is calculated to be 0.72. The value in this table at the intersection of 0.7 and x.x2 is
0.7642, which means that there is a 0.7642 (or 76.42%) probability of a value falling between negative
infinity and a Z score of 0.72.
Note that all probabilities for negative Z scores are below 50% in this table and all probabilities for positive
Z scores are above 50%. This is because the probability of falling between negative infinity and 0 (middle of
curve) for all normal curves is 50%.
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Z Table: Form 1
Z x.x0 x.x1 x.x2 x.x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9
-3.5 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002
-3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002
-3.3 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0004 .0004
-3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005
-3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007
-3.0 .0014 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010
-2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014
-2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019
-2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026
-2.6 .0047 .0045 .0044 .0043 .0042 .0040 .0039 .0038 .0037 .0036
-2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048
-2.4 .0082 .0080 .0078 .0076 .0073 .0071 .0069 .0068 .0066 .0064
-2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084
-2.2 .0139 .0136 .0132 .0129 .0126 .0122 .0119 .0116 .0113 .0110
-2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143
-2.0 .0227 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183
-1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233
-1.8 .0359 .0352 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294
-1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367
-1.6 .0548 .0537 .0526 .0515 .0505 .0495 .0485 .0475 .0465 .0455
-1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0570 .0559
-1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681
-1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823
-1.2 .1151 .1131 .1112 .1094 .1075 .1056 .1038 .1020 .1003 .0985
-1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170
-1.0 .1587 .1563 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379
-0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611
-0.8 .2119 .2090 .2061 .2033 .2004 .1977 .1949 .1921 .1894 .1867
-0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2207 .2177 .2148
-0.6 .2742 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451
-0.5 .3085 .3050 .3015 .2980 .2946 .2912 .2877 .2843 .2809 .2776
-0.4 .3446 .3409 .3372 .3336 .3300 .3263 .3228 .3192 .3156 .3121
-0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483
-0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859
-0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247
-0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641Vi lla
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Z Table: Form 1
z x.x0 x.x1 x.x2 x.x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9
+0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
+0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
+0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
+0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
+0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
+0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
+0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
+0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
+0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
+0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8304 .8365 .8389
+1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
+1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
+1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
+1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
+1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
+1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
+1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
+1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
+1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
+1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
+2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
+2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
+2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
+2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
+2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
+2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
+2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
+2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
+2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
+2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
+3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
+3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
+3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
+3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
+3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
+3.5 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998Vi lla
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Z Table: Form 2
Z Table: Form 2 The normalized Z table appears in three basic forms. The three basic forms are:
• Form 1 – Probability from negative infinity to a calculated Z value. Typical tabulated Z values
include -3.59 to +3.59 by increments of 0.01. This document can be found in a topic sheet for
form 1
• Form 2 – Probability from a Z value of 0.00 to a calculated Z value. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01
• Form 3 – Probability from a calculated Z value to positive infinity. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01. This document can be found in a topic sheet
for form 3
Regardless of the form used, a correct answer can be obtained when any of these three forms is properly used
to find normal probabilities. The following is an example of a Form 2 Z table which shows the probability
from a Z value of 0.00 to the calculated Z value. The Z value is calculated to 2 decimal places. The whole
number and first decimal place are found in the first column labeled “z” and the second decimal place is
found in one of the adjacent columns labeled “x.x0,” “x.x1,” x.x2,” etc. The value at the intersection of the
row and column selected is the area under the curve (probability) between negative infinity and that Z score.
Example: A Z score is calculated to be -2.45. This form of the Z table does not include any negative Z values,
but they are not needed because the Normal distribution is symmetric. This means that the probability of
falling a certain distance from the mean is the same in either direction. So the value in this table at the
intersection of 2.4 and x.x5 is 0.4929, which means that there is a 0.4929 (or 49.29%) probability of a
value falling between the mean of the distribution (Z = 0.00) and a calculated Z score of 2.45. This also
means that there is the same probability (49.29%) of falling between the mean and a Z score of -2.45.
Example: A Z score is calculated to be 0. The value in this table at the intersection of -0.0 and x.x0 is
0.0000, because the distance between the mean and itself is 0.
Example: A Z score is calculated to be 0.72. The value in this table at the intersection of 0.7 and x.x2 is
0.2642, which means that there is a 0.2642 (or = 26.42%) probability of a value falling between the mean
of the distribution and a Z score of 0.72.
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Z Table: Form 2
Z x.x0 x.x1 x.x2 x.x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
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Z Table: Form 3
Z Table: Form 3 The normalized Z table appears in three basic forms that are easily confused. The three basic forms are:
• Form 1 – Probability from negative infinity to a calculated Z value. Typical tabulated Z values
include -3.59 to +3.59 by increments of 0.01. This document can be found in a topic sheet for
form 1
• Form 2 – Probability from a Z value of 0.00 to a calculated Z value. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01. This document can be found in a topic sheet
for form 2
• Form 3 – Probability from a calculated Z value to positive infinity. Typical tabulated Z values
include from 0.00 to 3.09 by increments of 0.01
Regardless of the form used, a correct answer can be obtained when any of these three forms is properly used
to find normal probabilities. The following is an example of a Form 3 Z table which shows the probability
from positive infinity to the calculated Z value. The Z value is calculated to 2 decimal places. The whole
number and first decimal place are found in the first column labeled “z” and the second decimal place is
found in one of the adjacent columns labeled “x.x0,” “x.x1,” “x.x2,” etc. The value at the intersection of the
row and column selected is the area under the curve (probability) between negative infinity and that Z score.
Example: A Z score is calculated to be -2.45. This form of the Z table does not include negative Z values, but
they are not necessary since the Normal distribution is symmetric. The value in this table at the intersection
of 2.4 and x.x5 is 0.00714, which means that there is a 0.00714 (or .714%) probability of a value falling
between positive infinity and a Z score of 2.45. This also means that there is the same probability of falling
between negative infinity and a Z score of -2.45.
Example: A Z score is calculated to be 0. The value in this table at the intersection of -0.0 and x.x0 is .5000,
which means that there is a .5000 (or 50%) probability of a value falling between positive infinity and a Z
score of 0.
Example: A Z score is calculated to be 0.72. The value in this table at the intersection of 0.7 and x.x2 is
0.23576, which means that there is a 0.23576 (or multiplied by 100 = 23.576%) probability of a value
falling between positive infinity and a Z score of 0.72.
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Z Table: Form 3
Z x.x0 x.x1 x.x2 x.x3 x.x4 x.x5 x.x6 x.x7 x.x8 4.0 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002
3.9 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003
3.8 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005
3.7 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008
3.6 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012
3.5 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017
3.4 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025
3.3 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036
3.2 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052
3.1 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074
3.0 0.00135 0.00131 0.00128 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104
2.9 0.00187 0.00181 0.00175 0.00170 0.00164 0.00159 0.00154 0.00149 0.00144
2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199
2.7 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272
2.6 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368
2.5 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00509 0.00494
2.4 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657
2.3 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866
2.2 0.01390 0.01355 0.01321 0.01287 0.01255 0.01223 0.01191 0.01160 0.01130
2.1 0.01787 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463
2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876
1.9 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385
1.8 0.03593 0.03515 0.03438 0.03363 0.03288 0.03216 0.03144 0.03074 0.03005
1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754
1.6 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648
1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705
1.4 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944
1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08692 0.08534 0.08379
1.2 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10384 0.10204 0.10027
1.1 0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900
1.0 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007
0.9 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354
0.8 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19490 0.19215 0.18943
0.7 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770
0.6 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825
0.5 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096
0.4 0.34458 0.34090 0.33724 0.33360 0.32297 0.32636 0.32276 0.31918 0.31561
0.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197
0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974
0.1 0.46017 0.45621 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858
0.0 0.50000 0.49601 0.49292 0.48803 0.48405 0.48006 0.47608 0.47210 0.48612
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Type I and Type II Errors in Hypothesis Testing
Type I and Type II Errors in Hypothesis Testing There are four possible outcomes when making hypothesis test decisions from sample data. Two of these
outcomes are correct in that the sample accurately represents the population and leads to a correct
conclusion, and two are incorrect, as shown in the following figure:
THE TRUTH
The null hypothesis
(HO) is true
(Ha is false)
The null hypothesis
(H0) is not true
(Ha is true)
THE DECISION THE
ANALYST MAKES
Reject H0
(support Ha)
TYPE I (α) error/ Alpha Risk/ p – value
Overreacting
(1 - α) = the Confidence level of the test
Correct Decision (1 - β)
Power of the test
Fail to Reject H0
(do not support Ha)
Correct Decision TYPE II (β) error/ Beta Risk
Underreacting
TYPE I ERROR (or α Risk or Producer's Risk) In hypothesis testing terms, α risk is the risk of rejecting the null hypothesis when it is really true and
therefore should not be rejected. In other words, the alternative hypothesis is supported when there is
inadequate statistical evidence for doing so (too much risk). This can be thought of as overreacting to data
results that might be due just to chance alone.
The most commonly used level of α risk is .05, or 5%. This level of α risk means that there is a 5% chance
that the sample results are due to chance alone, so there is a 5% chance that rejecting the null hypothesis
(supporting the alternative hypothesis) will be an incorrect decision.
TYPE II ERROR (or β Risk or Consumer's Risk)
In hypothesis testing terms, β risk is the risk of failing to reject the null hypothesis when it is really false and
therefore should be rejected. In other words, the alternative hypothesis is not supported even though there
is adequate statistical evidence to show that supporting it meets the acceptable levels of risk. This can be
thought of as underreacting to data results that are probably real and not due just to chance alone.
The most commonly used level of β risk is .10, or 10%. This level of β risk means that there is a 10% chance
that the sample results are not due to chance alone, so there is a 10% chance that failing to reject the null
hypothesis (failing to support the alternative hypothesis) will be an incorrect decision.
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P-Values: Example 1
P-Values: Example 1 The only noticeable difference between using a software package for Hypothesis Testing and completing the
analyses manually is that manual testing establishes a target distance for the test statistic that will ensure the
desired level of α risk is maintained, while software will calculate the actual level of α risk. The results will be
the same either way since the targeted level of α risk should always be set prior to completing the analysis to
minimize the risk of settling for weak results.
The α risk is labeled p-value in the software printouts. For example, a p-value of 0.162 indicates that there is
a 16.2% risk that the test results are not statistically significant and are due instead to random chance (the
alternative hypothesis is false). A p-value of 0.005 indicates that there is only a 0.5% risk of the results not
really being significant.
Example: One sample t-test
Test of mu = 15 vs. not = 15
N Mean StDev SE Mean 95% CI T P
20 16.800 2.900 0.648 (15.443, 18.157) 2.78 0.012
This is a printout from Minitab™ software for a two-tailed t-test of the hypothesis that the population mean is
equal to 15.0 vs. the alternative hypothesis that the population mean is not equal to 15. Based on a sample
of 20 with a sample mean of 16.8 and sample standard deviation of 2.90, the software yields a p-value
of 0.012. This p-value suggests that the probability of a Type I error when you reject the null hypothesis is
0.012, or 1.2%. Since this is below the maximum alpha risk of 5% established for this analysis, the null
hypothesis should be rejected. The correct conclusion is that the population mean is not equal to 15.
Note: P-values are provided by software and are not generally calculated by hand.
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P-Values: Example 2
P-Values: Example 2 The only noticeable difference between using a software package for Hypothesis Testing and completing the
analyses manually is that manual testing establishes a target distance for the test statistic that will ensure the
desired level of α risk is maintained, while software will calculate the actual level of α risk. The results will be
the same either way since the targeted level of α risk should always be set prior to completing the analysis to
minimize the risk of settling for weak results.
The α risk is labeled p-value in the software printouts. For example, a p-value of 0.162 indicates that there
is a 16.2% risk that the test results are not statistically significant and are due instead to random chance (the
alternative hypothesis is false). A p-value of 0.005 indicates that there is only a 0.5% risk of the results not
really being significant.
Example: Two sample F test
Null hypothesis Sigma(C1)/Sigma(C2) = 1
Alternative hypothesis Sigma(C1)/Sigma(C2) not = 1
Variable N StDev Variance
C1 30 4.495 20.204
C2 30 7.411 54.927
Ratio of standard deviations = 0.606
Ratio of variances = 0.368
Method DF1 DF2 Statistic P-Value
F Test (normal) 29 29 0.37 0.009
This is a printout from Minitab™ software for a two-tailed test of the hypothesis that the population variances
are equal to each other or not equal to other. Based on a sample of 30 with a sample variance of 20.204
and a sample of 30 with a sample variance of 54.297, the software yields a p-value of 0.009. This p-value
suggests that the probability of a Type I error when you reject the null hypothesis is 0.009, or 0.9%. Since
this is below the maximum alpha risk of 5% established for this analysis, the null hypothesis should be
rejected. The correct conclusion is that the population variances are not equal to each other.
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P-Values: Example 3
P-Values: Example 3 The only noticeable difference between using a software package for Hypothesis Testing and completing the
analyses manually is that manual testing establishes a target distance for the test statistic that will ensure the
desired level of α risk is maintained, while software will calculate the actual level of α risk. The results will be
the same either way since the targeted level of α risk should always be set prior to completing the analysis to
minimize the risk of settling for weak results.
The α risk is labeled p-value in the software printouts. For example, a p-value of 0.162 indicates that there
is a 16.2% risk that the test results are not statistically significant and are due instead to random chance (the
alternative hypothesis is false). A p-value of 0.005 indicates that there is only a 0.5% risk of the results not
really being significant.
Example: Test for Normality
Below is a display from Minitab™ for a normality test on 30 data points. The null hypothesis in a test for
Normality is that the data fit a Normal distribution. The p-value is the probability of making a Type I error by
incorrectly rejecting the null hypothesis and concluding that the sample data was not taken from a normally
distributed dataset. In this example, the p value is 0.168 or 16.8%. Since this is beyond the maximum of 5%
typically used for alpha risk, the null hypothesis cannot be rejected, and we will continue to assume that the
data fit a Normal distribution.
Note: P-values are provided by software and are not generally calculated by hand.
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Student's t Distribution
Student’s t Distribution The Student’s t Distribution is closely related to the z distribution – it is the continuation of the z distribution
for cases when σ is unknown or when sample sizes are “small,” which is generally accepted to be below
about 30 pieces of data. Both z and t distributions are symmetric and bell-shaped, and both have a mean
of zero. The critical difference is that the t distribution is more variable due to the impact the smaller sample
sizes have on the standard deviation. The t distribution becomes narrower (taller) as sample sizes increase,
and gradually becomes very close to the Normal Distribution.
Key point: Evaluate “small” sample sizes, otherwise, to use the z distribution it is recommended that the
minimum pieces of data be greater than 30.
t Distributions must therefore be analyzed based on the amount of data available in the sample by calculating
the degrees of freedom (df) from the sample size. Degrees of freedom can be thought of as “computing or
estimating power” – larger sample sizes are more powerful for statistical analysis than smaller ones. The
formula for degrees of freedom is often (n – 1), but it varies with the analysis being done so the correct
formula should always be verified before use.
Example: The following illustrates the relationship between a normal z distribution (the solid black curve) and
the t distribution with 3 degrees of freedom (the dashed red line). As you can see, the t distribution is flatter
and wider than the z distribution.
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Student's t Distribution: Example
Student’s t Distribution: Example An order entry process was designed to average 4.5 minutes per application. A random sample of 12 orders
yielded the following data (in minutes):
5.2 4.7 4.3 5.5 4.5 4.5
5.5 4.5 5.3 4.7 5.1 4.8
Based on these data, is the average processing time for this process 4.5 minutes? Test at α = .05.
Solution:
= 4.88 minutes (sample mean) s = 0.42 minutes (sample standard deviation)
H0: µ = 4.5 minutes Ha: µ ≠ 4.5 minutes
Rejection region: For α = .05/2 = .025 and (n - 1) = (12 - 1) = 11, tα/2 = 2.201. The rejection region is
therefore any t beyond +/- 2.201. Since the t calculated from the sample data is 3.13, it does fall in the
rejection region (> 2.201), so the null hypothesis is rejected.
“Based on this study, I am 95% confident that the average processing time for these orders is statistically
significantly different from the target value of 4.5 minutes per order.”
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Student's t Distribution Table
Student’s t Distribution Table Values of the student’s t distribution are found in tabular form for selected values of alpha (α) for one-
tailed tests or alpha divided by 2 (α/2) for two-tailed tests. The correct t values depend upon the degrees
of freedom. The formulas for degrees of freedom are based on the sample size and vary with the test being
performed. The t values in this table are based on the area under the curve (probability) in the tail area of
the normal distribution being studied (α or α/2) and degrees of freedom for the study. For example, if you
are doing a one-tailed test with 5% alpha risk, you would use the 5% one-tailed column. If your degrees of
freedom are 11, the t value would then be 1.796.
One Tailed 10% 5% 2.5% 1% 0.5% 0.25% 0.1% 0.05%
Two Tailed 20% 10% 5% 2% 1% 0.5% 0.2% 0.1%
df
1 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6
2 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60
3 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92
4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869
6 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792
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Student's t Distribution Table
23 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.767
24 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725
26 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707
27 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690
28 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674
29 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659
30 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646 ∞ 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
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Paired t Test: Description
Paired t Test: Description Most hypothesis tests assume a random selection of the elements contained in each. This assumption does
not hold true, however, when two groups that are being compared are somehow related to one another.
Examples:
• The MEAN of pre-test scores compared to the MEAN of post-test scores
• The MEANS of two judges scoring a process
• Two same objects measured by two different measurement devices
When the sample groups are not independent, the appropriate method to use to test for differences between
the groups is known as a paired comparison test (or paired t-test or paired sample test). The null hypothesis
for a paired t-test is that the average difference between the two values in each pair of data is zero (0).
Example: Consider a situation in which a group of employees is tested both before and after a training
program to determine whether the training resulted in improved performance (the difference in the average
scores in the pre- and post-tests).
Employee Pre-Test Post-Test Differences
1 90 98 8
2 84 94 10
3 90 91 1
4 83 88 5
5 80 86 6
6 77 82 5
7 76 80 4
8 72 76 4
You need to know four (4) things:
1. The Mean of the old method/pre-test (µ): 81.5
2. The Mean of the new method/post-test ( ): 86.875
3. Standard deviation (SD): 2.7226
(calculated from the 8 differences – see table above)
4. n: 8 (in this particular case)
Now, right away you can see that the post-test scores have an average that is higher than the pre-test
scores. You might be tempted to conclude that the training is effective at improving employee performance.
It might APPEAR to be better, but the question is - is the performance found to be SIGNIFICANTLY BETTER
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Paired t Test: Description
than prior to the training? That is where the paired t-test comes into play. We are going to find out at some
significance level (e.g., 0.1, 0.05, or 0.01) whether or not the post test scores show significantly better
employee performance. These three levels of SIGNIFICANCE correspond to the 90%, 95%, or 99% levels of
CONFIDENCE, respectively.
How do you know whether to choose a 0.1, 0.05, or a 0.01 level of confidence? It depends on what risk of
being incorrect (alpha risk or beta risk) you are willing to live with. If you recall, the alpha risk is concluding
that something is significantly different or better – when it really IS NOT. The beta risk is opposite, where the
conclusion is that something is not significantly different, or better – when it really IS.
What we are looking at in this case, if we were to reject the null hypothesis, is an alpha risk. In other words,
if we conclude that the training resulting in significantly better performance, there is a risk that we are wrong
– even though the math is correct. If it turns out that we are wrong, we will have made a Type I error.
NOTE: A Type II error is tied to the beta risk.
In this example, let's say the team decided on a 0.05 level of significance. (This corresponds to a 95% level
of confidence.)
Here is the paired t-test formula:
The next question the team needs to ask is whether this test is targeting a larger-is-better, a smaller-is-better,
or a nominal-is-best quality characteristic? The reason we need to know this is so that we can determine
whether this is a one-tail test, or a two-tail test. In this case, we are trying to determine whether the training
has resulted in significantly better performance (i.e., larger test scores), so this is a larger-is-best quality
characteristic. This means that this is a one-tailed test because we are only concerned with the right tail of
the curve closest to the 100% mark for test scores. If, on the other hand, we were only concerned whether or
not it made a difference either way, then it would be a two-tailed test and we have to account for both tails of
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Paired t Test: Description
the curve and split the alpha into two halves. If we were looking for a smaller-is-better characteristic, (which
we are not) we would be looking only at the left side of the curve and in that case it would be a one-tailed
test.
We now need to look up the table value – the critical value to determine whether or not the MEAN of the test
scores after the training (the test statistic of 5.5838) is significantly better (larger) than the MEAN of the test
scores prior to training.
We need to determine the degrees of freedom (df). When dealing with the t-test, we take n-1 to find the
critical value in the t-distribution table. So, in this case, we had eight (8) pairs of data which means that the
degrees of freedom is seven (7).
The team had decided on a 95% level of confidence. This corresponds with the 5% level of significance.
Refer to the table (below) and where the 5% (level of significance) column intersects with the role for seven
(7) degrees of freedom, the critical value is 1.895. This means that if our calculated TEST STATISTIC is
beyond (or larger than) the CRITICAL VALUE of 1.895, then we can conclude that we have rejected the null
hypothesis. Since our calculated value of 5.5838 IS beyond the critical value of 1.895, we DO reject the null
hypothesis. What does this mean? It means we can conclude with a 95% level of confidence that the new
training method makes an improvement (towards a larger-is-better characteristic) in the student’s knowledge.
Remember, we could be wrong. There is a 5% probability that we have concluded that the new training
program has made a significant impact on employee performance when in fact it might not be the case. The
team should be prepared to live with that small probability of being wrong. That risk is known as the alpha
risk. If it turns out that the team was that wrong, they will have made a Type I error.
0.1 (10%) 0.05 (5%) 0.01 (1%)
df = 1 3.078 6.314 31.82
df = 2 1.886 2.920 6.965
df = 3 1.638 2.353 4.541
df = 4 1.533 2.132 3.747
df = 5 1.476 2.015 3.365
df = 6 1.440 1.943 3.143
df = 7 1.415 1.895 2.998
df = 8 1.397 1.860 2.896
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Paired t Test: Student's t Distribution Table
Paired t Test: Student’s t Distribution Table
One Tailed 10% 5% 2.5% 1% 0.5% 0.25% 0.1% 0.05%
Two Tailed 20% 10% 5% 2% 1% 0.5% 0.2% 0.1%
df
1 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6
2 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60
3 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92
4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869
6 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792
23 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.767
24 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725
26 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707
27 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690
28 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674
29 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659
30 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646
40 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551
50 1.299 1.676 2.009 2.403 2.678 2.937 3.261 3.496
60 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460
80 1.292 1.664 1.990 2.374 2.639 2.887 3.195 3.416
∞ 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
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x-bar and R Control Chart Construction Rules
x-bar and R Control Chart Construction Rules 1. The data must be continuous.
2. The sample size (sub-group) must be at least 2 and should not be more than about 10–12. (If the sample
size is larger than about 10–12, the x-bar and s chart should be used to get a more accurate estimate of
the process variation.)
3. The sample size cannot vary.
4. Collect the data in a random and consecutive manner. (In production order)
5. For the first chart on a process, collect at least 20–25 sample groups (subgroups) to ensure that an
adequate measure of process variation has been taken.
6. Calculate the data points to be plotted on the chart.
• : Average of data values for each time period
• R: Range of values in the data set for each time period
7. Calculate the centerline for the chart. Note: The centerline is always the average of all the plotted data
points in the start-up group.
• (“x double bar”): The average of the subgroup averages calculated from the baseline chart data
• (“R bar”): The average of the subgroup ranges calculated from the baseline chart data
8. Calculate the 3-sigma control limits for the chart using the appropriate formula and constant factors as
shown in the following tables:
Chart Type Upper Control Limit Lower Control Limit
x-bar (with Range) + A2 - A2 Range D4 D3
n A2 D3 D4
2 1.88 0 3.27
3 1.02 0 2.57
4 0.73 0 2.28
5 0.58 0 2.11
6 0.48 0 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
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x-bar and R Chart: Example
x-bar and R Chart: Example The following is an example of how the control limits are computed for an x-bar and R chart. Note that at
least 25 sample subgroups should be used to get an accurate measure of the process variation. The subgroup
sample size used here is 3, but it can range from 2 to about 10–12 and is typically around 5.
Sample 1 Sample 2 Sample 3 Sample 4 Sample 5
11.1 10.1 9.8 11.3 11.2
9.2 11.2 10.2 10.1 9.4
11.3 9.9 9.9 10.1 8.9
x-bar 10.5 10.4 10.0 10.5 9.8
R 2.1 1.3 0.4 1.2 2.3
Note: D3, D4, and A2 were all obtained from the Control Chart Constants Table for a sample size of n = 3.
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Measurement Discrimination
Measurement Discrimination If a basketball coach measured the heights of the players to the nearest yard, all of the players would be two
yards tall. The conclusion would be that there is no variation in the heights of basketball players. However, if
the coach measured all of the players to the nearest foot, the players would either be 6 feet tall or 7 feet tall.
Measure to the nearest inch, and the result would be a large variation in the heights of basketball players.
How fine a measurement is sliced is referred to as Measurement Discrimination.
One way to visualize the amount of measurement discrimination is to look at the range portion of a control
chart. It is points (dots) that you need to look at on the chart. You look for the number of strata (parallel
layers) of the dots as they spread across the range chart. It is not the count of dots. Instead, it is a count of
strata (parallel layers) of dots (you are looking for a minimum of six strata).
If given these ranges:
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
If all of the range values are at zero, there is only one strata – the zero line.
But if given these ranges:
0.0 0.5 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.5 0.0 0.0 0.5
Dots are found on the zero line and the 0.5 line. You would have two strata.
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Measurement Discrimination
And if given these ranges:
0.0 0.5 0.0 0.0 1.0 0.0 1.0
1.0 0.0 0.0 0.5 0.0 0.0 0.5
You could find dots on the zero line, the 0.5 line, and the 1.0 line.
There would be three strata.
Finally, if you had these ranges:
0.5 0.0 2.0 1.0 1.0 2.0 0.0
1.0 2.0 0.5 1.0 2.0 0.0 1.0
You can see dots on the zero line, the 0.5 line, the 1.0 line, and the 2.0 line, but none of the dots on the 1.5
line. You still have five strata, even though you do not see any of the dots on the 1.5 line. There could be a
dot on the 1.5 line – and there would eventually be – if we were to continue to gather data.
If you are using a control chart to determine measurement discrimination, the rule-of-thumb is to have at least
six strata (units of measure) under the upper control limit of the range. Note: the zero line (strata) counts as
one of the six desired units.
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Measurement Discrimination: Why Six Points?
Measurement Discrimination: Why Six Points? Why six points? To understand why it is six points we are looking for, there needs to be an understanding of rounding. When
measurements are rounded off too much, most of the information about dispersion is lost.
Example:
Individual values
72.7136
72.8248
72.1241
72.0136
72.6141
72.5098
72.3132
72.0144
If we had rounded (tens place), we would have gotten the following measurements:
70
70
70
70
70
70
70
70
We would have concluded: zero dispersion
If we had rounded (ones place), we would have gotten the following measurements:
73
73
72
72
73
73
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Measurement Discrimination: Why Six Points?
72
We would have concluded: very little dispersion. Possible range values: 0, and the one’s place value. (Units
= 2)
If we had rounded (tenths place), we would have gotten the following measurements:
72.7
72.8
72.1
72.0
72.6
72.5
72.3
72.0
We would have concluded: some dispersion. Possible range values: 0, and the one’s place value, and the
tenth’s place holder. (Units = 3)
If we had rounded (hundredth place), we would have gotten the following measurements:
72.71
72.82
72.12
72.01
72.61
72.51
72.31
72.01
We would have concluded: more about dispersion. Possible range values: 0, and the one’s place value, the
tenth’s place holder, and the hundredths place holder. (Units = 4)
Four is better than three. Five is better than four, and so on.
Considering standard deviation in this example is 0.320 * 1.128 = 0.361 is the average range. Our spread
of dispersion is ~ 0.361 units, yet our measure increments are less than that. Our measurement increments
are in varying values of 0.01 which is less than our dispersion. It stands to reason that if we want to be able
to detect variation, we need measurement increments that are less than standard deviation; else we cannot
detect it. For example, if the only means for detecting the weights of humans is measured to the nearest ton,
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Measurement Discrimination: Why Six Points?
“I Can’t Believe It’s Not Butter®” It is a little-known fact that “I Can’t Believe It’s Not Butter®” suggests there is zero fat in their yellow spray
bottle, when in fact there is fat in that bottle – and plenty of it.* The food producer is taking advantage of the
F.D.A.’s rules for rounding that measures grams of fat per serving down to the whole number. Since one squirt
is the serving size (5 sprays for topping), and since five squirts is less than one half of a gram of fat, they
round down to zero, and that is exactly what they put on the label.
*13 sprays is about 1 gram of fat, 25 sprays is about 2 grams of fat. 38 sprays is about 3 grams of fat. The
whole bottle contains 90 grams of fat. So if you can’t believe it’s not butter, pour it on and watch the scale.
Why 6 units? It has to do with the discrimination ratio. If you want more on this, read the book by Dr. Donald Wheeler,
Evaluating the Measurement System, (2nd Ed.). Wheeler states,
“Since this problem [indiscriminate measurement] arises out of the inability to detect variation with the
subgroups, the solution consists of increasing the ability to detect that variation. This can be done in one
of two ways. Either use smaller measurement units, or increase the variation within the subgroups to a
detectable level. The latter approach will usually be accomplished by collecting values for a subgroup over a
longer time span. By increasing the time span, one often increases the variation enough to make it detectable
with the original measurement units.”
It is referred to in the course as a larger inference space.
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Rational Subgroup
Rational Subgroup Rational Subgrouping is the name given to the way in which data are organized into subgroups for process
control charts. Rational subgroups for process control charts involve the use of process and product
knowledge and judgment, but there are a few basic principles that should be followed.
• Usually, the subgroups should be chosen so that within subgroup variation is minimized. For
example, keep in mind that for an Average and Range chart, it is the within subgroup variation
that determines how sensitive the chart will be since the average Range (R-bar) is used in the
calculation of the Averages chart control limits
• The selection of rational subgroups should be consistent with the structure of the data from the
process
• The selection of rational subgroups should allow for quick identification of potential corrective
actions once an out-of-control condition is identified
Example: A process involves the filling of one-gallon cans of an expensive liquid using a machine with three
filling heads. The operator has taken some initial data on the process and is trying to decide how to organize
it into a process control chart. The initial data is given below. The characteristic being measured is fill volume
(in gallons).
The operator wants to obtain rational
subgroups for the Average and Range
chart but isn’t sure what that is. After
consulting with a quality engineer, the
following options emerge.
Option 1: Organize the data into subgroups of three taken each hour. The problem is that this approach will
mix together the variation of the filling heads and make an out-of-control point on the average chart difficult to
interpret.
Option 2: Organize the data by filling head utilizing an Individuals and Moving Range chart and plotting all
three heads on the same chart. Again, this will be difficult to interpret. What will an out-of-control point
mean? You would somehow need to identify each head on the chart.
Option 3: Treat each measure of each head as a rational subgroup of one and run three Individuals and
Moving Range Charts (taking a measure from each head each hour). While this requires the use of three
charts, it is the most productive organization of the data since it allows for quick interpretation of an out-of-
control condition.
Filling head
8 a.m. 9 a.m. 10 a.m. 11 a.m. 12 noon
1 1.02 1.04 1.00 1.01 0.99
2 1.03 1.04 1.01 1.02 1.03
3 1.02 1.02 1.00 1.01 1.03
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Control Chart Construction: Formulas for Centerlines
Control Chart Construction: Formulas for Centerlines The following formulas are used to compute the Centerlines for Statistical Process Control (SPC) charts:
Control Chart Type Centerline Symbol
Individual The average of the single (individual) data values for the
baseline chart
(x bar)
Moving Range The average of the moving (or rolling) ranges calculated from
the chart data
X-bar The average of the subgroup averages calculated from the
baseline chart data (x double bar)
Range The average of the subgroup ranges calculated from the
baseline chart data
np The average of the np’s (number good/bad) in each sample
group collected for the chart (np bar)
p The average of the p’s (proportion good/bad) calculated for
each sample group in the chart (p bar)
c The average of the c’s (total number of flaws, defects,
occurrences, etc.) in each sample group collected for the
chart
(c bar)
u The average of the u’s (average number of flaws, defects,
occurrences, etc., per unit) calculated for each sample group
(u bar)
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Control Chart Construction: Formulas for Control Limits
Control Chart Construction: Formulas for Control Limits The following formulas are used to compute the Upper and Lower Control Limits for Statistical Process Control
(SPC) charts. Values for A2, A3, B3, B4, D3, and D4 are all found in a table of Control Chart Constants.
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Control Chart Tests: Shewhart's Test for Outliers
Control Chart Tests: Shewhart’s Test for Outliers A chart fails for outliers if more than about 0.3% of the plotted points fall outside of the 3 sigma control
limits. (Note that since outliers are very rare, most people investigate them all, regardless of the total number
beyond the control limits.)
Why? About 99.7% of the data points in a normal distribution will fall within + / -3 standard deviations from
the average – the remaining 0.3% will naturally fall outside. Since this is a very small number, plotted points
beyond the control limits should rarely occur.
Possible causes of outliers include:
• The control limits or outliers have been calculated or plotted incorrectly
• Something in the measuring system changed temporarily, such as a new gage, a new inspector or
operator, an inconsistent or unclear operational definition, etc.
• A sudden, significant, and temporary special cause is present, such as a vacation operator, a temporary
procedure change, an equipment failure, a new advertising campaign, a holiday or special event, etc.
EXAMPLE: This chart fails with 2 outliers.
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Control Chart Tests: Shewhart's Test for a Sudden and Drastic Shift in Level
Control Chart Tests: Shewhart’s Test for a Sudden and Drastic Shift in Level A chart fails for a sudden and drastic shift in level if a group of plotted points suddenly jumps to a higher or
lower group level (average) than the preceding points. A group is generally considered to be at least 5 or more
points, and a sudden jump is generally considered to be at least 1 standard deviation band.
Why? The average of all of the plotted points should remain the same throughout the chart.
Possible causes of runs include:
• The process has shifted to a new average due to a sudden and at least temporarily sustained change in
some aspect of the process
EXAMPLE: This chart fails twice for a sudden and drastic shift in level because a group of 6 points suddenly
jumps to a different average (level) in the +2 sigma band, and then a group of at least 6 points suddenly
jumps to a different average (level) in the -1 sigma band.
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Control Chart Tests: Shewhart's Test for a Trend/Gradual Change in Level
Control Chart Tests: Shewhart’s Test for a Trend/Gradual Change in Level A chart fails a trend/gradual change in level (average) if a long series of plotted points gradually moves toward
either control limit, or if there is a long series of data points without a change in direction.
Why? A trend indicates the process average is increasing or decreasing and no longer at the centerline of
the control chart. Stable processes will display data that are horizontal over time – the average will not be
changing.
Possible causes of a trend include:
• Causes which work on the process gradually, such as inadequate maintenance of equipment, operator
fatigue, operator learning, temperature changes, etc.
EXAMPLE: This chart fails for a trend because the points are gradually approaching the upper control limit.
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Failure Mode and Effects Analysis (FMEA)
Failure Mode and Effects Analysis (FMEA) Failure Mode and Effects Analysis (FMEA) is a risk management tool that is often used in quality and
reliability engineering to identify high-risk items based on the consequences of failure. An FMEA can be a
process FMEA (where the risks are process failures) or a design FMEA (where the risks are product or system
related failures). To assess risk, an FMEA utilizes the product of three measures (frequency of occurrence,
severity of consequence, and chance of detection) to create a risk priority number (RPN) which functions as
a numerical assessment of the risk of that failure mode. The tool is sometimes referred to as Failure Modes,
Effects, and Criticality Analysis or FMECA.
Example: A professional association is concerned about the process it uses to issue membership cards. They
decide to conduct an FMEA of the process and part of the FMEA is given below.
Mode of Failure
Cause Effect Frequency Severity Detection RPN
Card printed incorrectly
Incorrect information provided
Card must be reissued
3 8 5 120
Information incorrectly entered in database
Card must be reissued
5 8 5 200
The risk priority number (RPN) is the product of the frequency, severity, and detection values. The frequency
value represents how likely this cause is to occur; the severity value indicates how significant the impact is;
and the detection value measures how likely this mistake is to be detected internally. In this example, the
RPN value indicates the cause “information incorrectly entered into database” should be given priority for
corrective action.
Keep in mind that most companies either develop their own measures of the three categories (frequency,
severity, and detection) or they utilize standard values for their industry. Moreover, for an FMEA to be effective
the categories must have reliable measures and there must be consensus regarding the value of the severity
column that should trigger automatic corrective action.
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Poka-Yoke: Part 1
Poka-Yoke: Part 1 Poka-yoke (poh ka yoke) is a mistake-proofing approach to eliminating errors that was developed by the
Japanese engineer Shigeo Shingo in the 1960s. The word “poka-yoke” is derived from a Japanese word that
means “to avoid errors.” Poka-yoke makes use of simple but effective tools and signals to prevent errors from
occurring.
Example 1: The widespread use of ATMs has certainly made banking more convenient and shows no signs
of diminishing usage. However, early on, the banks encountered a very inconvenient problem; customers
regularly drove off from the ATM leaving their debit cards in the card slot of the ATM. A simple but effective
poka-yoke has been incorporated in the ATMs so that an alarm sounds and continues to sound until you
have taken your card from the slot. This type of poka-yoke can be thought of as an encounter error-proofing
example since it intervenes in the service encounter and attempts to prevent a mistake (in this case, a
mistake by the customer).
Example 2: A very common poka-yoke is involved every time we purchase fuel for our automobiles. When
unleaded fuel was introduced, there were many automobiles on the road that still used leaded fuel. In order to
prevent leaded fuel from being placed in unleaded automobiles, the size of the fuel inlet was changed so that
the leaded fuel dispense nozzle would not insert.
Again, you can see that “error-proofing” or poka-yoke attempts to prevent an error from occurring.
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Poka-Yoke: Part 2
Poka-Yoke: Part 2 Example 3: A small assembler provides wiring harness assemblies for a particular automotive electronic
controller. There are several part numbers and many of the parts for each part number look very much alike.
So it isn’t surprising that the company has received complaints from the automotive assembly plants that
some harnesses have incorrect components (which have to be changed out on the assembly line causing
delays). Their solution is to color code the different part numbers (green for one part number and yellow
for the other – see the illustration below) and to make sure that only the containers with the appropriate
part number make it to the work stations. The color coding is vintage poke-yoke and successfully prevented
incorrect components from being assembled into the wiring harnesses. See the illustration below.
As you can see, poka-yoke makes extensive use of the philosophy of visual control by using visual cues to
assist in the prevention of errors.
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Poka-Yoke: Part 3
Poka-Yoke: Part 3 Example 4: The service world has recognized the value of poka-yoke during the last several years. One way
of applying error-proofing (poka-yoke) in the service sector is to devise a “task” poka-yoke. A task poka-yoke
concentrates on the tasks that a server must perform and devises methods to assure the task is completed
correctly. A popular example of a task poka-yoke is the automatic change dispenser that you encounter in
many stores. The cash register automatically dispenses the correct change once the purchased item and the
payment have been entered. This prevents an error in the “task” of making change.
Example 5: A CPA firm has experienced delays in preparing tax returns because the customer hasn’t provided
all the information required. The firm decides to use a “preparation” poka-yoke. A preparation poka-yoke is an
intervention in the service encounter that attempts to properly prepare the customer for the service. The CPA
firm mails out a check list to each of their customers that contains a list and a place to check it off for each
item critical to the preparation of that customer’s tax return. See the example below.
Required Item Check
1 Previous year’s tax return.
2 Organized receipts for tax deductions.
3 All W-2s.
4 Mortgage information.
5 Healthcare expenses.
Example 6: A small manufacturer of printed signs has received complaints from its customers regarding print
imperfections. Until now the firm has required inspectors to inspect every item of the printed sign (like color
variation, printing streaks, etc.). After discussing the inspection protocol with the inspectors, the firm decided
that it was asking too much of the inspectors by insisting that they inspect so many items. Together with the
inspection personnel, and using the complaints received, they compiled a short list of the most important
inspection items. To ensure the inspectors did not forget to inspect carefully these items, they created an
inspection board that contained lights by each of the critical items. The inspector must press a button to turn
the light off, thereby assuring the inspector has looked carefully at the critical item.
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Chi-Square: Example
Chi-Square: Example The Barnes Company manufactures a DVD player and claims that the mean number of hours of use before
repairs are needed is 400, with a standard deviation of 10 hours. The specified variance, therefore, is σo2
= 102 = 100 hours2. A new company marketing representative suspects that the “before repair” variance
is actually less than 100 hours2. To verify this, she tests nine machines and finds a sample mean of 410
hours and a standard deviation of 5.5. Is the sample variance statistically significantly less than the currently
claimed variance? Use α = 0.05.
Solution: Since it must be determined whether σ2 < 100, the elements of the test are:
H0: σ2 = 100 Ha: σ2 < 100
Test Statistics:
Rejection region: For α = 0.05 and (n - 1) = (9 - 1) = 8 degrees of freedom, σ2(1-α)= 2.73 because this
will yield the point beyond which there is at most a 5% chance of a sample variance falling if the population
variance is equal to 100.
Since the X2 value calculated from the sample data is 2.42, it falls in the rejection region (<2.73), so the
null hypothesis is rejected. This means that the marketing representative has sufficient evidence to conclude
with 95% confidence that the actual variance is less than the 100 hours2 currently claimed by the Barnes
Company.
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Chi-Square: Table
Chi-Square: Table This version of the X2 table is based on the area under the curve (probability) between 0 and the point of
interest along the horizontal axis. The degrees of freedom (ν) is found using the formula appropriate for the
analyses being performed. For example, if you are doing a 5% one-tailed, less-than test with 19 degrees of
freedom, the X2 table value would be 10.1.
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Chi-Square Distribution: ν = 1
Chi-Square Distribution: ν= 1 The chi-square probability density function for 1 degree of freedom is illustrated below. It is skewed to the
right, and since the random variable on which it is based is squared, it has no negative values.
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Chi-Square Distribution: ν = 5
Chi-Square Distribution: ν = 5 The chi-square probability density function for 5 degrees of freedom is illustrated below. It is skewed to the
right, and since the random variable on which it is based is squared, it has no negative values.
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Chi-Square Distribution: ν = 10
Chi-Square Distribution: ν = 10 The chi-square probability density function for 10 degrees of freedom is illustrated below. It is skewed to the
right, and since the random variable on which it is based is squared, it has no negative values. As the degrees
of freedom increases, the probability density function begins to appear symmetrical in shape. This distribution
plot has been enhanced to show the a 0.025 probability space in each tail and the associated chi-square
values (3.427 and 20.48).
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Chi-Square Distribution: ν = 15
Chi-Square Distribution: ν = 15 The chi-square probability density function for 15 degrees of freedom is illustrated below. As the degrees of
freedom increases, the probability density function begins to appear symmetrical in shape.
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F Distribution
F Distribution The F Distribution is closely related to the Chi-Square Distribution in that it is based on the relationship (ratio)
between two independent Chi-Square random variables. As such, the F distribution can never have negative
values, and it ranges from zero to positive infinity. The F distribution is used in testing the relationship
between two population variances with sample data, and in Designed Experiments using Analysis of Variance
(ANOVA).
Following is a graph of the F distribution given degrees of freedom for the numerator (df1) of 9 and the
degrees of freedom for the denominator of 19. Values for the F distribution are usually given in probability
tables. The F values are found in the table based on the degrees of freedom for the numerator and
denominator of the sample statistics.
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F Test for Variance: Example
F Test for Variance: Example A comparison of the precision of two machines developed for extracting juices from oranges is to be made
using the following data:
MACHINE 1: s12 = 3.1 oz2 n1 = 25
MACHINE 2: s22 = 1.4 oz2 n2 = 25
Is there sufficient evidence to indicate that the variance of Machine 1 is greater than the variance of Machine
2? Test at α = .05.
Solution: Let population 1 be the population of measurements on Machine 1. (This is done to make this an
upper-tailed test by placing the larger variance in the numerator.)
Rejection Region: Since the two sample sizes are the same, ν1 = ν 2 = (25 - 1) = 24. This means that
F.05,24,24 = 1.98 because this will yield the ratio of population variances beyond which there is at most a 5%
chance of a ratio of sample variances falling if the two population variances are equal. Since F = 2.21 >
1.98, the null hypothesis is rejected, and it is concluded that the variability of Machine 1 is greater than the
variability of Machine 2.
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ANOVA – One Way
ANOVA – One Way A One-Way ANOVA is a statistical method for comparing the effect of the levels of a single factor on a
response variable. If each level is of the same sample size, then the design is called balanced; otherwise it is
unbalanced.
Example: Suppose you are interested in comparing the productivity (as measured in volume generated) of
four machines that produce machine castings. Over the course of a month, you randomly select nine days
of production for each machine (and record the number of castings produced that day) which yields the data
below. Note this is a balanced design.
The procedure works as follows: The null
hypothesis is that all levels have equal average
productivity, while the alternative hypothesis is
that at least one of the groups (levels) has an
average productivity significantly different from
the others. An F statistic is calculated from the
sample data (an F statistic is the ratio of the
Mean Square for Treatment or Between Groups
with the Mean Square for Error or Within
Groups).
If the calculated F statistic is greater than the appropriate value of the critical F (found in a table or provided
in software), then the null hypothesis is rejected. A one-way ANOVA generated by Excel (with alpha equal to
0.05) is given below.
ANOVA – One Way
Source of Variation SS df MS F P-value F crit
Between Groups 15641.89 3 5213.96 3.19 0.037 2.90
Within Groups 52285.11 32 1633.91
Total 67927.00 35
Since the computed F statistic (3.19) is greater than the Critical F (2.90), we reject the null hypothesis and
conclude that at least one of the machines has an average productivity significantly different from the others
(at an alpha level of 0.05). Note the p-value is given (0.037) and it is below 0.05. Note also that there is
nothing special about the alpha level of 0.05. Depending upon the ramifications of a wrong conclusion, the
team may have chosen a 0.1 or a 0.01 alpha level.
Machine 1 Machine 2 Machine 3 Machine 4
404 423 436 454
381 436 369 483
394 393 452 500
461 413 531 442
407 447 354 393
396 369 440 475
432 495 428 480
393 390 435 505
443 441 351 464
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Analysis of Variance (ANOVA): One-Way Manufacturing Example A two-sample t-test will allow us to compare or test the equality of means for two samples, but when we
move beyond two samples, we need another method. At first it might seem that we should conduct multiple
two-sample tests until we have conducted all possible two-sample comparisons, but this approach has
practical and theoretical problems. The most serious issue, actually, is the theoretical one. Conducting
multiple two-sample comparisons causes the probability of falsely rejecting a true null hypothesis to increase
as the number of t-tests increases. Instead, we use one of the most often used methods in all of statistics: the
Analysis of Variance (ANOVA).
The Analysis of Variance results in a single test of the hypothesis that all group means are equal. That is, the
hypotheses for the One-Way ANOVA are:
Ho: µ1 = µ2 = µ3 = … = µn Ha: at least one group mean is different
ANOVA also requires the following conditions be met:
• Each of the n populations under consideration has a normal distribution
• The variances (σ2) of the n populations are approximately equal
• The measurements represent independent random samples from their respective populations
One of the simplest experiments we can conduct is a one way comparison. Suppose we have three adhesives
(A, B, and C) and we want to compare the average break-away force (in grams) when applied to a material
used in a specialty product. The data are given below.
Adhesive A Adhesive B Adhesive C
203 325 307
315 434 329
373 196 287
351 309 416
409 337 339
375 374 335
359 478 375
385 486 301
Average 346.25 367.38 336.13
Variance 4,094.79 9,469.70 1,778.13
Analysis of Variance (ANOVA): One-Way Manufacturing Example Vi
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Since the adhesives differ significantly in price, we would like to know if the adhesives differ in resulting
break-away force. We apply each adhesive to a random sample of 8 production material surfaces (essentially
the same material in each case). The Analysis of Variance works as follows: we generate two estimates of the
variance (one generated from “between” groups and another generated “within” groups). Also, the variances
aren’t actually equal, but they are close enough for the application of ANOVA.
First we compute the “sum of squares” between using the following formula (where k is the number of groups
and j is the jth group):
Note that the average of the group averages (x-bar-bar) is equal to 349.92. So, the sum of squares (between)
is:
SS(Between)=8(346.25 – 349.92)2 + 8(367.38 – 349.92)2 + 8(336.13 – 349.92)2
=107.56 + 2,438.35 + 1,521.68 = 4,067.59
Next we compute the “mean square” between using the following formula (where k is the number of groups):
Now we have an estimate from “between” groups and so we move to compute an estimate for “within” groups
(again, k is the number of groups and j is the jth group).
Thus, the sum of squares within is
SS(Within)= (8-1)4,094.79 + (8-1)9,469.70 + (8-1)1,778.13 = 107,398
The “mean square” within is (where nT is the total number of data values)
The Analysis of Variance test significance using the F distribution (which is the distribution used to model the
ratio of two variances). The idea (developed by one of the most famous statisticians, R. A. Fisher, and thus
the F test) is that the ratio of the two estimates of variance cannot be too large without the group means being
different in some way. We compute the F statistic as follows:
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Excel’s “ANOVA Single Factor” produces the following:
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 4,067.58 2 2,033.79 0.40 0.677 3.47
Within Groups 107,398.25 21 5,114.20
Total 111,465.83 23
The F statistic has two “degrees of freedom,” a numerator degree of freedom and a denominator degree
of freedom. So, in this example, the “critical” or theoretical value of F would have k – 1 = 2 numerator
degrees of freedom and nT – k = 21 denominator degrees of freedom. Suppose we choose an alpha level of
0.05. Then the critical value of F for alpha = 0.05, 2 numerator degrees of freedom, and 21 denominator
degrees of freedom is 3.47 (Excel lists this value at the end of the ANOVA output or see an F table). Since our
computed value of F is 0.40 which is NOT greater than the critical value of F (which is 3.47 – see F table),
we do NOT reject the null hypothesis.Therefore, we cannot conclude that the type of adhesive has an impact
on the average break-away force.
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Analysis of Variance (ANOVA): One-Way Service Example A two-sample t-test will allow us to compare or test the equality of means for two samples, but when we
move beyond two samples, we need another method. At first it might seem that we should conduct multiple
two-sample tests until we have conducted all possible two-sample comparisons, but this approach has
practical and theoretical problems. The most serious issue, actually, is the theoretical one. Conducting
multiple two-sample comparisons causes the probability of falsely rejecting a true null hypothesis to increase
as the number of t-tests increases. Instead, we use one of the most often used methods in all of statistics: the
Analysis of Variance (ANOVA).
The Analysis of Variance results in a single test of the hypothesis that all group means are equal. That is, the
hypotheses for the One-Way ANOVA are:
Ho: µ1 = µ2 = µ3 = … = µn Ha: at least one group mean is different
ANOVA also requires the following conditions be met:
• Each of the n populations under consideration has a normal distribution
• The variances (σ2) of the n populations are approximately equal
• The measurements represent independent random samples from their respective populations.
Suppose we want to investigate the possibility that the average salaries for clerical assistants are different
among three offices. We obtain a simple random sample of six clerical assistants from each of the three
offices. The data from the samples are given below.
Office 1 Office 2 Office 3
$28,726 $21,597 $26,290
$23,095 $27,267 $20,763
$25,183 $28,825 $17,174
$22,337 $29,722 $18,327
$21,180 $24,230 $17,482
$19,124 $29,478 $19,966
Average $23,274.17 $26,853.17 $20,000.33
Variance 11,177,186.17 10,737,526.17 11,454,590.67
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A boxplot comparison (from Minitab) yields the following. The dark circle is the average in each case (the
circle with the crosshair is the median).
There is overlap between the boxplots, so we need ANOVA to help us decide. The Analysis of Variance works
as follows: we generate two estimates of the variance (one generated from “between” groups and another
generated “within” groups). Also, the variances aren’t actually equal, but they are close enough for the
application of ANOVA.
First we compute the “sum of squares” between using the following formula (where k is the number of groups
and j is the jth group):
Note that the average of the group averages (x-bar-bar) is equal to $23,375.89. So, the sum of squares
(between) is:
SS(Between) = 6(23,247.17 – 23,375.89)2 + 6(26,853.17 – 23,375.89)2 + 6(20,000.33 – 23,375.89)2
=62,084.46 + 72,548,764.46 + 68,366,251.85 = 140,977,101
Next, we compute the “mean square” between using the following formula (where k is the number of groups):
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Now we have an estimate from “between” groups; so we can compute an estimate for “within” groups (again,
k is the number of groups and j is the jth group).
Thus, the sum of squares within is
SS(Within) = (6–1)11,177,186.17 + (6–1)10,737,526.17 + (6–1)11,454,590.67 = 166,846,515
The “mean square” within is (where nT is the total number of data values)
The Analysis of Variance test significance using the F distribution (which is the distribution used to model the
ratio of two variances). The idea (developed by one of the most famous statisticians, R. A. Fisher, and thus
the F test) is that the ratio of the two estimates of variance cannot be too large without the group means
being different in some way. We compute the F statistic as follows:
Excel’s “ANOVA Single Factor” produces the following:
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 140,977,100.8 2 70,488,550.39 6.34 0.0101 3.68
Within Groups 166,846,515.0 15 11,123,101.00
Total 307,823,615.8 17
The F statistic has two “degrees of freedom” – a numerator degree of freedom and a denominator degree of
freedom. So, in this example, the “critical” or theoretical value of F would have k – 1 = 2 numerator degrees
of freedom and nT – k = 15 denominator degrees of freedom. Suppose we choose an alpha level of 0.05.
Then the critical value of F for alpha = 0.05, 2 numerator degrees of freedom, and 15 denominator degrees
of freedom is 3.68 (Excel lists this value at the end of the ANOVA output or see F table). Since our computed
value of F is 6.34 which is greater than the critical value of F (which is 3.68 – see F table), we reject the
null hypothesis and conclude at least one of the groups has an average significantly different from the others
(Office 2).
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Design of Experiments (DOE) Terminology: Designed Experiment
Design of Experiments (DOE) Terminology: Designed Experiment The following claim isn’t an overstatement: To a great degree designed experiments and the methods of
experimental design are the very essence of the scientific method. Some famous experimental design experts
have said it this way:
Scientific research is a process of guided learning. The object of statistical methods is to make that process as
efficient as possible.1
As the principles and practices of Six Sigma have become more widely applied, more and more people
are beginning to understand the importance of the scientific method as a “learning guide” for quality
improvement. Think about the basic Plan → Do → Study → Act cycle. Now consider the basic flow of the
scientific method.
We always begin with observations. Something causes us to form
questions or to confront problems with performance, or quality, or
expectations. From the observations and questions, we develop
hypotheses about causes and relationships. At this point, however,
our views are no more than mere hypotheses. Before these views
can be converted to “knowledge” so that learning occurs, they must
be “tested” – they must systematically confront empirical conditions
that allow for an efficient assessment of their value. Only then can we
knowingly accept, reject, or modify them. This process is at the heart of
experimental design.
Douglas Montgomery defines an experiment as follows: “a test or series of tests in which purposeful changes
are made to the input variables of a process or system so that we may observe and identify the reasons for
changes in the output response.”2 There are several key words in this definition.
• Purposeful changes: in a designed experiment, the “design” is important. We intend to make systematic
changes so that we can assess the impact of these changes. Without careful attention to the design of
tests, we will not be able to confidently evaluate the extent to which different variables are contributing to
the observed response.
1 Statistics for Experimenters, 1st. Ed., G. E. P. Box, W. G. Hunter, and J. S. Hunter, John Wiley & Sons, Inc., 1978, page 1. 2 Design and Analysis of Experiments, 4th Ed., John Wiley & Sons, Inc., 1997, page 1.
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• Process or system: we are never experimenting in a vacuum. We are working in the context of a process
or system and it is critically important that we understand the key drivers of the performance of that
process or system.
• Output response: the response variable is the “score-keeper” of our experiment, since it is how we will
measure and analyze the results. Therefore, it is vital that we understand the response, that we can show
it is a valid measure, and that we can measure it reliably. Too much measurement error has ruined many
well-intentioned experiments.
Thus, when we refer to a designed experiment, we are talking about a process of learning and investigation
with (at least) the following characteristics:
1. A clearly defined and valid response variable that can be measured reliably and accurately.
2. A systematic plan or series of tests that allow for an effective and efficient analysis of the results of our
experiment.
3. Input variables (or factors) that are related to the response variable in some way (or, at least, that we
think are related).
As an illustration, consider a very basic experiment. Suppose we have four different methods of training new
customer service technicians. We would like to compare these four methods and, hopefully, decide which
one is “best.” Of course, we need a definition of “best” in order to compare the methods. So, further suppose
that we have developed several “virtual situations” that we can present and we have an expert panel that
will score the customer service technician’s response to the virtual situations. Our test design is as follows:
randomly assign n = 20 new customer service technician’s to the four training methods (five to each method)
and then let the expert panel score each technician’s handling of each virtual situation. The response will be
the total score (with a higher score being better). The table below contains the results of our tests.
It certainly isn’t obvious whether or no there is a
real difference in the training methods. We need a
way of analyzing the data so that we can compare
the different methods and test for a significant
difference. Typically, the statistical method known as
One-way Analysis of Variance would be used here.
A One-Way ANOVA (you can use Excel for this) shows there is no significant difference in the training methods
based on the experts’ scores of the customer service technicians (see the ANOVA Table below).
One-Way ANOVA
Source of Variation SS df MS F P-value
Between Groups 28.00 3.00 9.33 0.70 0.565
Within Groups 212.80 16.00 13.30
Total 240.8 19
Method 1 Method 2 Method 3 Method 4
32 34 27 33
25 31 36 32
32 32 29 35
32 26 29 38
38 32 30 29
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The p-value that results from the comparison is 0.565 – way above any acceptable value (for example, 0.05
is often used). Therefore, we cannot reject the hypothesis that there is no difference in the training methods.
While this might be disappointing, it is also important. That is, at least we aren’t tempted to spend money
adopting a particular method that, in fact, shows no real improvement in training.
While the above experiment is relatively simple, it demonstrates some of the basic principles of experimental
design and it also presents some questions that commonly arise. For example:
• Are we sure the experts are measuring the trainees consistently?
• Are the virtual situations realistic and comprehensive?
• Are the training methods really that different?
• Did we test enough technicians?
Finally, keep in mind that a good experimental design allows the experimenter to deal effectively with three
common challenges.3
• The influence of experimental error. Variation occurs in every measurement and we need to be able
to partition this error in a way that allows us to identify the impact on the response attributable to the
factors.
• The temptation to confuse correlation with causation. Just because two variables are correlated doesn’t
mean they are causally related. They might both be associated with a third variable. The process of
randomization in experimental design can help reduce the likelihood that we make this mistake.
• The complexity that might arise from multiple effects. It is tempting to quickly conclude that the impact
on the response is just the added impact of the input variables. Unfortunately, this is often not the case.
The impact on the response could be the result of an interaction. Only good experimental design will
allows us to identify this.
3 Statistics for Experimenters, 1st. Ed., G. E. P. Box, W. G. Hunter, and J. S. Hunter, John Wiley & Sons, Inc., 1978, page 7.
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Design of Experiments (DOE) Terminology: Response Variable
Design of Experiments (DOE) Terminology: Response Variable The response variable in a designed experiment is the variable we are measuring as a result of the
experimental trials. It is the outcome of our experiment and its values will drive our analysis of the impact
of the factors. Consequently, it is crucial that we be able to measure the response variable reliably and
accurately. Essentially, this means that we have conducted a measurement systems analysis and that the
measurement error associated with our attempt to estimate the response is acceptable.
It is possible for the response variable to be either quantitative or qualitative. However, the most useful
response variables in experimental design tend to be quantitative. The kinds of response variables that might
prove useful are almost endless. Below are a few examples.
• In manufacturing situations, it is quite common to have a response variable like diameter, thickness,
smoothness, and so on
• In service industries, many response variables are related to cycle time or to customer satisfaction
measures
Again, the key is to make sure the response variable is clearly defined and that it can be reliably measured.
Consider the scenario where a large provider of training services is concerned about a new training program
targeting financial analysts. The company would like to know when the analysts prefer the training and if the
session works better with one or two trainers. In this case the response variable is likely to be the rating given
for the training program. That is, a summary of the experimental variables might look like the following.
Time of Presentation
# of Presenters
Response Variable
Morning or One or Presentation
Afternoon Two Rating
An experimental design must be chosen that will allow them to measure the impact of each of these two
“factors” at their respective “levels” on the response variable “Presentation Rating.” Of course, they would
want to be sure that they can reliably measure “Presentation Rating” or the experiment could be a waste of
time.
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Design of Experiments (DOE) Terminology: Factors and Levels
Design of Experiments (DOE) Terminology: Factors and Levels The factors in a designed experiment are the parameters we vary in order to induce changes in the response
variable. The idea is to “control” these factors by setting them at chosen levels and using an experimental
design that will allow us to measure the impact of the factor at each specific level on the response.
Factors are typically chosen in the experimental design process by having the design of experiments team
brainstorm a list of factors believed to be related to the response variable. The brainstorming session usually
generates more factors than you have time to investigate, so the team will need to reduce the list to a
manageable size. Once the team has reached a consensus about the subgroup of factors to be included in the
experiment, the next step is choose the levels at which those factors will be tested.
Levels are the specific settings that we choose for each factor. This is often a difficult task. For very practical
reasons, it is advisable (if possible) to restrict the levels to two or three. The reason for this, again, is very
practical: the more levels you choose for a given factor, the larger the experimental design will be (and the
more costly and complex it will become). However, there are times when the experimental situation will
demand a factor at four or five levels.
Consider the following situation. A design of experiments team has chosen three factors to include in an
experimental design investigating the bond strength of a cover used in an electronic assembly. They chose
as their factors: Material Type (two options), Adhesive Type (two kinds), and Curing Time (two times). The
response will be the force required to separate the cover. The factor and levels for the experiment can be
summarized as follows.
Factors Level 1 Level 2
Material Type Material A Material B
Adhesive Type Adhesive 1 Adhesive 2
Curing Time 24 hours 48 hours
Note that the first two factors are “qualitative” (different types of things) while the third factor is quantitative
(time allowed to cure). The ability to accommodate both kinds of factors is one of the great advantages of
experimental design.
Keep in mind that choosing factor levels is an art. For quantitative factor levels, you want to choose them
far enough apart to generate a detectable difference on the response, but not so far apart that it isn’t useful.
Also, note that when choosing two levels for a factor, you are implicitly assuming the behavior of the response
between the two levels is linear. If you doubt this, check it out first.
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Control Plan
Control Plan Description: A Control Plan is a method for documenting the functional elements of quality control that are
to be implemented in order to assure that quality standards are met for a particular product or service. The
intent of the control plan is to formalize and document the system of control that will be utilized.
Example: A company is preparing to transition into production a redesigned, small electric motor. One of the
critical parts of the motor is the metal shaft. A portion of a control plan for the machining of the shaft is given
below.
Control Item Characteristic Gage Specification Frequency Chart Respon.
Shaft Length Laser 10cm, ± 0.1 3 per hr X-bar, R Operator
Typically, the control plan may also include other items like: the frequency with which the process is reviewed,
verification that the measurement system is capable, typical corrective actions to be taken in the presence
of out-of-control conditions, any special inspection requirements, and a history of the process capability
measures.
Note that the control plan is a basic working document for quality engineering (in both manufacturing and
service). One should be able to review the control plan and quickly understand the elements of quality
assurance being utilized.
And remember, this example is a manufacturing example. The same technique can be applied to any process
(e.g., services industry, back-office, transactional processes). The team should use a control plan to ensure
that the process does not slip back into the ways of the old process.
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Standard Operating Procedures
Standard Operating Procedures A Standard Operating Procedure (SOP) is a set of written instructions that document a routine activity that
is to be followed by members of an organization. Standard Operating Procedures are essential parts of good
quality systems. Sound and well-written SOPs should enhance consistency and reduce human error. Note:
sometimes standard operating procedures are referred to as Work Instructions.
Example: An example of what is becoming a common standard operating procedure can be seen in almost
any food service organization or hospital: washing hands. Most of the hand-washing SOPs contain some
version of the following.
Inadequate SOP: 1. Wet hands and forearms.
2. Thoroughly apply the soap.
3. Lather the hands and forearms.
4. Continue to lather and scrub.
5. Thoroughly rinse forearms and hands.
6. Dry hands and forearms thoroughly.
This SOP lacks detail.
Better: 1. Wet hands and forearms with warm, running water (the water temperature should reach at
least 100 ºF)
2. Thoroughly apply the soap from the forearms to the hands.
3. Lather the hands and forearms. Be sure to get soap under fingernails and between fingers.
4. Continue to lather and scrub for at least 10 seconds.
5. Thoroughly rinse forearms and hands with clean water. No remaining soap should be visible on
the forearms, hands, or under nails.
6. Dry hands and forearms thoroughly with single-use paper towels.
7. Use the paper towel to turn off the water and use the same paper towel to open the door as
you exit the restroom.
This is a better SOP because it provides more specificity.
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Standard Operating Procedures
Best: 1. Wet hands and forearms with warm, running water (the water temperature should reach at least
100 ºF)
2. Thoroughly apply the soap to the hands.
3. Lather the hands and forearms. Be sure to get soap under fingernails and between fingers.
4. Continue to lather and scrub for at least 10 seconds.
5. Thoroughly rinse forearms and hands with clean water. No remaining soap should be visible on
the forearms, hands, between finders, or under nails.
6. Dry hands and forearms thoroughly with single-use paper towel.
7. Use the paper towel to turn off the water and use the same paper towel to
open the door as you exit the restroom. If there is no waste basket in the
washing area, carry the paper towel to a place that has one, as long as you do
not have to touch anything between the washing room and the process.
Providing pictures or illustrations take away some of the ambiguity.
SOP Effectiveness: Creating a work instruction is only the beginning. What good is an SOP if it isn’t being followed? It is
recommended that periodic audits are conducted to ensure that the SOPs are being followed fully and
consistently over time.
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Cp and Cpk: Formulas
Cp and Cpk: Formulas The Cp and Cpk capability indices estimate the performance of the process if all assignable causes of
variation (all shifts, drifts, etc.) can be removed from the process.
The Cp and Cpk indices use the average range to estimate the process variation with the following formula:
The d2 factor is based on sample size and found in the following table:
SAMPLE SIZE
d2 SAMPLE
SIZE d2
SAMPLE SIZE
d2
2 1.128 9 2.970 16 3.532
3 1.693 10 3.078 17 3.588
4 2.059 11 3.173 18 3.640
5 2.326 12 3.258 19 3.689
6 2.534 13 3.336 20 3.735
7 2.704 14 3.407 25 3.931
8 2.847 15 3.472 30 4.086
Since the average range is a measure of the variation within each sample subgroup, is known as the within
variation.
The Cp compares the total predicted process variation (defined as +/- 3 standard deviations) to the allowable
process variation (specification range):
The Cpk compares the actual process center and spread to the nominal or target process center and spread:
Cpk is based on the distance from the process mean to the nearest, and therefore riskiest, performance target
so the smallest value is always selected.
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Cp and Cpk: Example
Cp and Cpk: Example An order-entry process is required to be performed on an average of 7 + / - 2 minutes. A control chart was
placed on the process and 25 sample groups of 5 items each were collected. It was found that the process
was stable and normally distributed with an average time of 7 minutes and an average range of 3.101
minutes.
A Cp of 0.5 means that the tolerance interval is half the width of the total predicted process variation (or the
process variation is twice the size of the tolerance interval). This means that this process will have a large
percentage of orders entered outside of the target time frame even if the average processing time is in the
center of the target interval. A Cpk of 0.5 means that only half of a normal curve will fit between the average
and the nearest performance limit.
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- Classic Wastes
- Prioritization Matrix
- Brainstorming
- Cause and Effect Diagram
- Affinity Diagram
- Tree Diagram
- Interrelationship Diagram
- Process Decision Program Charts (PDPC)
- Activity Network Diagram
- Project Charter: Description
- Project Charter: Blank Worksheet
- Project Charter: Example
- Project Charter Highlights
- Critical-to-Quality Tree
- Flow Chart
- Swim Lane Flow Chart
- Basic Statistics Using the Texas Instruments TI-30 Calculator
- Standard Deviation: 6 Steps to Calculation
- Standard Deviation: Description and Example
- Histograms
- Cell Intervals: Impact on Shape
- Cell Intervals: Rules of Thumb
- Stem-and-Leaf Plots
- Box and Whiskers Plot
- Check Sheets: Defect Location Check Sheet
- Run Chart: Basic Construction
- Run Chart: Shifts
- Run Chart: Trends
- Target Values: Illustration 1
- Target Values: Illustration 2
- Target Values: Illustration 3
- DPU, DPMO, PPM, and RTY
- DPMO to Sigma Level Relationship
- Pp and Ppk: Formulas
- Pp and Ppk: Example
- Finding a Z Score: Example 1
- Finding a Z Score: Example 2
- Finding a Z Score: Example 3
- Z Table: Form 1
- Z Table: Form 2
- Z Table: Form 3
- Type I and Type II Errors in Hypothesis Testing
- P-Values: Example 1
- P-Values: Example 2
- P-Values: Example 3
- Student's t Distribution
- Student's t Distribution: Example
- Student's t Distribution Table
- Paired t Test: Description
- Paired t Test: Student's t Distribution Table
- x-bar and R Control Chart Construction Rules
- x-bar and R Chart: Example
- Measurement Discrimination
- Measurement Discrimination: Why Six Points?
- Rational Subgroup
- Control Chart Construction: Formulas for Centerlines
- Control Chart Construction: Formulas for Control Limits
- Control Chart Tests: Shewhart's Test for Outliers
- Control Chart Tests: Shewhart's Test for a Sudden and Drastic Shift in Level
- Control Chart Tests: Shewhart's Test for a Trend/Gradual Change in Level
- Failure Mode and Effects Analysis (FMEA)
- Poka-Yoke: Part 1
- Poka-Yoke: Part 2
- Poka-Yoke: Part 3
- Chi-Square: Example
- Chi-Square: Table
- Chi-Square Distribution: ν = 1
- Chi-Square Distribution: ν = 5
- Chi-Square Distribution: ν = 10
- Chi-Square Distribution: ν = 15
- F Distribution
- F Test for Variance: Example
- ANOVA – One Way
- Analysis of Variance (ANOVA): One-Way Manufacturing Example
- Analysis of Variance (ANOVA): One-Way Service Example
- Design of Experiments (DOE) Terminology: Designed Experiment
- Design of Experiments (DOE) Terminology: Response Variable
- Design of Experiments (DOE) Terminology: Factors and Levels
- Control Plan
- Standard Operating Procedures
- Cp and Cpk: Formulas
- Cp and Cpk: Example
- Blank Page
