six sigma case study?

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Lecture 2

Review of

basic statistics

IMPROVEMENT ROADMAP
Uses of Probability Distributions

Establish baseline data characteristics.

Project Uses

Identify and isolate sources of variation.

Use the concept of shift & drift to establish project expectations.

Demonstrate before and after results are not random chance.

Breakthrough

Strategy

Phase 4:

Control

Characterization

Phase 1:

Measurement

Phase 2:

Analysis

Optimization

Phase 3:

Improvement

Measurements are critical...

If we can’t accurately measure something, we really don’t know much about it.
If we don’t know much about it, we can’t control it.
If we can’t control it, we are at the mercy of chance.

Types of Data

Data where the metric is composed of a classification in one of two (or more) categories is called Attribute data. This data is usually presented as a “count” or “percent”.
Good/Bad
Yes/No
Hit/Miss etc.
Data where the metric consists of a number which indicates a precise value is called Variable data.
Time
Miles/Hr

Use Minitab to have students record the results and have the students display using Graph..Histogram
Note how “rough” the graph looks
Redo using Basic Statistics …. Descriptive Statistics and display using the Graphical Summary. Walk through the normal curve transform:
Mean (Arithmetic Average)
Standard Deviation
Skew (How off center the data is skewed -=left)
Kurtosis (How flat or peaked the data is -=flat)
Show the Box Plot:
Quartile (25% of the Data Points)
Median (50% of the Data Points on Each Side)
Show the 95% Confidence Interval and Explain how it relates to the data.

Probability and Statistics

Probability and Statistics influence our lives daily
Statistics is the universal language for science
Statistics is the art of collecting, classifying,

presenting, interpreting and analyzing numerical

data, as well as making conclusions about the

system from which the data was obtained.

Population Vs. Sample (Certainty Vs. Uncertainty)

A sample is just a subset of all possible values

population

sample

Since the sample does not contain all the possible values, there is some uncertainty about the population.
Hence any statistics, such as mean and standard deviation, are just estimates of the true population parameters.

Descriptive Statistics

Descriptive Statistics is the branch of statistics which

most people are familiar. It characterizes and summarizes

the most prominent features of a given set of data (means,

medians, standard deviations, percentiles, graphs, tables

and charts.

Descriptive Statistics describe the elements of

a population as a whole or to describe data that represent

just a sample of elements from the entire population

Inferential Statistics

Inferential Statistics is the branch of statistics that deals with

drawing conclusions about a population based on information

obtained from a sample drawn from that population.

While descriptive statistics has been taught for centuries,

inferential statistics is a relatively new phenomenon having

its roots in the 20th century.

We “infer” something about a population when only information

from a sample is known.

Probability is the link between

Descriptive and Inferential Statistics

USES OF PROBABILITY DISTRIBUTIONS

Primarily these distributions are used to test for significant differences in data sets.

To be classified as significant, the actual measured value must exceed a critical value. The critical value is tabular value determined by the probability distribution and the risk of error. This risk of error is called a risk and indicates the probability of this value occurring naturally. So, an a risk of .05 (5%) means that this critical value will be exceeded by a random occurrence less than 5% of the time.

Critical Value

Common Occurrence

Rare Occurrence

WHAT IS THE MEAN?

The mean is the most common measure of central tendency for a population. The mean is simply the average value of the data.

n=12

Mean

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

mean

WHAT IS THE MEDIAN?

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

If we rank order (descending or ascending) the data set for this distribution we could represent central tendency by the order of the data points.

If we find the value half way (50%) through the data points, we have another way of representing central tendency. This is called the median value.

Median Value

Median

50% of data points

WHAT IS THE MODE?

If we rank order (descending or ascending) the data set for this distribution we find several ways we can represent central tendency.

We find that a single value occurs more often than any other. Since we know that there is a higher chance of this occurrence in the middle of the distribution, we can use this feature as an indicator of central tendency. This is called the mode.

Mode

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

MEASURES OF CENTRAL TENDENCY, SUMMARY

MEAN ( )

(Otherwise known as the average)

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

n/2=6

}

Mode = 0

MEDIAN

(50 percentile data point)

Here the median value falls between two zero values and therefore is zero. If the values were say 2 and 3 instead, the median would be 2.5.

MODE

(Most common value in the data set)

The mode in this case is 0 with 5 occurrences within this data.

Median

n=12

SO WHAT’S THE REAL DIFFERENCE?

MEAN

The mean is the most consistently accurate measure of central tendency, but is more difficult to calculate than the other measures.

MEDIAN AND MODE

The median and mode are both very easy to determine. That’s the good news….The bad news is that both are more susceptible to bias than the mean.

SO WHAT’S THE BOTTOM LINE?

MEAN

Use on all occasions unless a circumstance prohibits its use.

MEDIAN AND MODE

Only use if you cannot use mean.

Example of tossing a coin 200 time
(probability of getting heads)

Number of occurrences

What are some of the ways that we can easily indicate the dispersion (spread) characteristic of the population?

Three measures have historically been used; the range, the standard deviation and the variance.

WHAT IS THE RANGE?

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

Range

MAX

MIN

(

)

The range is a very common metric which is easily determined from any ordered sample. To calculate the range simply subtract the minimum value in the sample from the maximum value.

Range

Max

Min

WHAT IS THE VARIANCE & STANDARD DEVIATION?

The variance (s2) is a very robust metric which requires a fair amount of work to determine. The standard deviation(s) is the square root of the variance and is the most commonly used measure of dispersion for larger sample sizes.

DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

-5-(-.17)=-4.83

-3-(-.17)=-2.83

-1-(-.17)=-.83

0-(-.17)=.17

1-(-.17)=1.17

3-(-.17)=3.17

4-(-.17)=4.17

(-4.83)2=23.32

(-2.83)2=8.01

(-.83)2=.69

(.17)2=.03

(1.17)2=1.37

(3.17)2=10.05

(4.17)2=17.39

61.67

(

)

-.17

MEASURES OF DISPERSION

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

ORDERED DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

Min=-5

max

min

(

)

Max=4

DATA SET

-5

-3

-1

-6

-5

-4

-3

-2

-1

-.17

(

)

-5-(-.17)=-4.83

-3-(-.17)=-2.83

-1-(-.17)=-.83

0-(-.17)=.17

1-(-.17)=1.17

3-(-.17)=3.17

4-(-.17)=4.17

(-4.83)2=23.32

(-2.83)2=8.01

(-.83)2=.69

(.17)2=.03

(1.17)2=1.37

(3.17)2=10.05

(4.17)2=17.39

61.67

RANGE (R)

(The maximum data value minus the minimum)

VARIANCE (s2)

(Squared deviations around the center point)

STANDARD DEVIATION (s)

(Absolute deviation around the center point)

SO WHAT’S THE REAL DIFFERENCE?

VARIANCE/ STANDARD DEVIATION

The standard deviation is the most consistently accurate measure of central tendency for a single population. The variance has the added benefit of being additive over multiple populations. Both are difficult and time consuming to calculate.

RANGE

The range is very easy to determine. That’s the good news….The bad news is that it is very susceptible to bias.

SO WHAT’S THE BOTTOM LINE?

VARIANCE/ STANDARD DEVIATION

Best used when you have enough samples (>10).

RANGE

Good for small samples (10 or less).

SO WHAT IS THIS SHIFT & DRIFT STUFF...

The project is progressing well and you wrap it up. 6 months later you are surprised to find that the population has taken a shift.

-12

USL

LSL

-10

-8

-6

-4

-2

SO WHAT HAPPENED?

Time

All of our work was focused in a narrow time frame. Over time, other long term influences come and go which move the population and change some of its characteristics. This is called shift and drift.

Historically, this shift and drift primarily impacts the position of the mean and shifts it 1.5 s from it’s original position.

Original Study

VARIATION FAMILIES

Variation is present upon repeat measurements within the same sample.

Variation is present upon measurements of different samples collected within a short time frame.

Variation is present upon measurements collected with a significant amount of time between samples.

Sources of Variation

Within Individual Sample

Piece to Piece

Time to Time

SO WHAT DOES IT MEAN?

To compensate for these long term variations, we must consider two sets of metrics. Short term metrics are those which typically are associated with our work. Long term metrics take the short term metric data and degrade it by an average of 1.5s.

IMPACT OF 1.5s SHIFT AND DRIFT

PPM

(+1.5

)

0.0

500,000

0.0

933,193

0.1

460,172

0.0

919,243

0.2

420,740

0.1

903,199

0.3

382,089

0.1

884,930

0.4

344,578

0.1

864,334

0.5

308,538

0.2

841,345

0.6

274,253

0.2

815,940

0.7

241,964

0.2

788,145

0.8

211,855

0.3

758,036

0.9

184,060

0.3

725,747

1.0

158,655

0.3

691,462

1.1

135,666

0.4

655,422

1.2

115,070

0.4

617,911

1.3

96,801

0.4

579,260

1.4

80,757

0.5

539,828

1.5

66,807

0.5

500,000

1.6

54,799

0.5

460,172

1.7

44,565

0.6

420,740

Here, you can see that the impact of this concept is potentially very significant. In the short term, we have driven the defect rate down to 54,800 ppm and can expect to see occasional long term ppm to be as bad as 460,000 ppm.

IMPROVEMENT ROADMAP
Uses of Probability Distributions

Breakthrough

Strategy

Phase 4:

Control

Characterization

Phase 1:

Measurement

Phase 2:

Analysis

Optimization

Phase 3:

Improvement

Baselining Processes

Verifying Improvements

Common Uses

Data points vary, but as the data accumulates, it forms a distribution which occurs naturally.

Location

Spread

Shape

Distributions can vary in:

PROBABILITY DISTRIBUTIONS,

WHERE DO THEY COME FROM?

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Sheet: Sheet5

Sheet: Sheet6

Sheet: Sheet7

Sheet: Sheet8

Sheet: Sheet9

Sheet: Sheet10

Sheet: Sheet11

Sheet: Sheet12

Sheet: Sheet13

Sheet: Sheet14

Sheet: Sheet15

Sheet: Sheet16

COMMON PROBABILITY DISTRIBUTIONS

-4

-3

-2

-1

-4

-3

-2

-1

Original Population

Subgroup Average

Subgroup Variance (s2)

Continuous Distribution

Normal Distribution

c2 Distribution

Population and Sample Symbology

Value

Population

Sample

Mean

Variance

Standard Deviation

Process Capability

Binomial Mean

Z TRANSFORM

-1s

+1s

68.26%

2 tail = 32%

1 tail = 16%

+/- 1s = 68%

-2s

+2s

95.46%

2 tail = 4.6%

1 tail = 2.3%

+/- 2s = 95%

-3s

+3s

99.73%

2 tail = 0.3%

1 tail = .15%

+/- 3s = 99.7%

Common Test Values

Z(1.6) = 5.5% (1 tail a=.05)

Z(2.0) = 2.5% (2 tail a=.05)

The Focus of Six Sigma…..

Y = f(x)

All critical characteristics (Y) are driven by factors (x) which are “downstream” from the results….

Attempting to manage results (Y) only causes increased costs due to rework, test and inspection…

Understanding and controlling the causative factors (x) is the real key to high quality at low cost...

Probability distributions identify sources of causative factors (x). These can be identified and verified by testing which shows their significant effects against the backdrop of random noise.

HOW DO POPULATIONS INTERACT?

ADDING TWO POPULATIONS

mnew

snew

Population means interact in a simple intuitive manner.

Means Add

m1 + m2 = mnew

Population dispersions interact in an additive manner

Variations Add

s12 + s22 = snew2

HOW DO POPULATIONS INTERACT?

SUBTRACTING TWO POPULATIONS

mnew

snew

Population means interact in a simple intuitive manner.

Means Subtract

m1 - m2 = mnew

Population dispersions interact in an additive manner

Variations Add

s12 + s22 = snew2

TRANSACTIONAL EXAMPLE

Orders are coming in with the following characteristics:

Shipments are going out with the following characteristics:

Assuming nothing changes, what percent of the time will shipments exceed orders?

X = $53,000/week

s = $8,000

X = $60,000/week

s = $5,000

TRANSACTIONAL EXAMPLE

shipments

orders

shipments

orders

$60

$53

000

(

)

(

)

shipments

orders

shipments

orders

5000

8000

$9434

$7000

Shipments > orders

X = $53,000 in orders/week

s = $8,000

X = $60,000 shipped/week

s = $5,000

Orders

Shipments

To solve this problem, we must create a new distribution to model the situation posed in the problem. Since we are looking for shipments to exceed orders, the resulting distribution is created as follows:

The new distribution looks like this with a mean of $7000 and a standard deviation of $9434. This distribution represents the occurrences of shipments exceeding orders. To answer the original question (shipments>orders) we look for $0 on this new distribution. Any occurrence to the right of this point will represent shipments > orders. So, we need to calculate the percent of the curve that exists to the right of $0.

TRANSACTIONAL EXAMPLE, CONTINUED

shipments

orders

shipments

orders

$60

$53

000

(

)

(

)

shipments

orders

shipments

orders

5000

8000

$9434

To calculate the percent of the curve to the right of $0 we need to convert the difference between the $0 point and $7000 into sigma intervals. Since we know every $9434 interval from the mean is one sigma, we can calculate this position as follows:

$7000

$9434

Look up .74s in the normal table and you will find .77. Therefore, the answer to the original question is that 77% of the time, shipments will exceed orders.

$7000

Shipments > orders

CORRELATION ANALYSIS

Correlation Analysis is necessary to:

show a relationship between two variables. This also sets the stage for potential cause and effect.

IMPROVEMENT ROADMAP
Uses of Correlation Analysis

Determine and quantify the relationship between factors (x) and output characteristics (Y)..

Common Uses

Breakthrough

Strategy

Phase 4:

Control

Characterization

Phase 1:

Measurement

Phase 2:

Analysis

Optimization

Phase 3:

Improvement

KEYS TO SUCCESS

Always plot the data

Remember: Correlation does not always imply cause & effect

Use correlation as a follow up to the Fishbone Diagram

Keep it simple and do not let the tool take on a life of its own

WHAT IS CORRELATION?

Input or x variable (independent)

Output or y variable (dependent)

Correlation

Y= f(x)

As the input variable changes, there is an influence or bias on the output variable.

A measurable relationship between two variable data characteristics.

Not necessarily Cause & Effect (Y=f(x))

Correlation requires paired data sets (ie (Y1,x1), (Y2,x2), etc)

The input variable is called the independent variable (x or KPIV) since it is independent of any other constraints

The output variable is called the dependent variable (Y or KPOV) since it is (theoretically) dependent on the value of x.

The coefficient of linear correlation “r” is the measure of the strength of the relationship.

The square of “r” is the percent of the response (Y) which is related to the input (x).

WHAT IS CORRELATION?

TYPES OF CORRELATION

Strong

Weak

None

Positive

Negative

Y=f(x)

CALCULATING “r”
Coefficient of Linear Correlation

Calculate sample covariance ( )
Calculate sx and sy for each data set
Use the calculated values to compute rCALC.
Add a + for positive correlation and - for a negative correlation.

(

)

(

)

CALC

While this is the most precise method to calculate Pearson’s r, there is an easier way to come up with a fairly close approximation...

APPROXIMATING “r”
Coefficient of Linear Correlation

Y=f(x)

+ = positive slope

- = negative slope

6.7

12.6

Plot the data on orthogonal axis
Draw an Oval around the data
Measure the length and width of the Oval
Calculate the coefficient of linear correlation (r) based on the formulas below

HOW DO I KNOW WHEN I HAVE CORRELATION?

The answer should strike a familiar cord at this point… We have confidence (95%) that we have correlation when |rCALC|> rCRIT.
Since sample size is a key determinate of rCRIT we need to use a table to determine the correct rCRIT given the number of ordered pairs which comprise the complete data set.
So, in the preceding example we had 60 ordered pairs of data and we computed a rCALC of -.47. Using the table at the left we determine that the rCRIT value for 60 is .26.
Comparing |rCALC|> rCRIT we get .47 > .26. Therefore the calculated value exceeds the minimum critical value required for significance.
Conclusion: We are 95% confident that the observed correlation is significant.

Ordered

Pairs

CRIT

.88

.81

.75

.71

.67

.63

.51

.44

.40

.36

.28

.22

100

.20

CENTRAL LIMIT THEOREM

For this module you will need 12 dice and flip charts.

The Central Limit Theorem is:

the key theoretical link between the normal distribution and sampling distributions.
the means by which almost any sampling distribution, no matter how irregular, can be approximated by a normal distribution if the sample size is large enough.

IMPROVEMENT ROADMAP
Uses of the Central Limit Theorem

Common Uses

Breakthrough

Strategy

Phase 4:

Control

Characterization

Phase 1:

Measurement

Phase 2:

Analysis

Optimization

Phase 3:

Improvement

The Central Limit Theorem underlies all statistic techniques which rely on normality as a fundamental assumption

WHAT IS THE CENTRAL LIMIT THEOREM?

Central Limit Theorem

For almost all populations, the sampling distribution of the mean can be approximated closely by a normal distribution, provided the sample size is sufficiently large.

Normal

What this means is that no matter what kind of distribution we sample, if the sample size is big enough, the distribution for the mean is approximately normal.

This is the key link that allows us to use much of the inferential statistics we have been working with so far.

This is the reason that only a few probability distributions (Z, t and c2) have such broad application.

If a random event happens a great many times, the average results are likely to be predictable.

Jacob Bernoulli

HOW DOES THIS WORK?

As you average a larger and larger number of samples, you can see how the original sampled population is transformed..

Parent Population

n=2

n=5

n=10

ANOTHER PRACTICAL ASPECT

This formula is for the standard error of the mean.

What that means in layman's terms is that this formula is the prime driver of the error term of the mean. Reducing this error term has a direct impact on improving the precision of our estimate of the mean.

The practical aspect of all this is that if you want to improve the precision of any test, increase the sample size.

So, if you want to reduce measurement error (for example) to determine a better estimate of a true value, increase the sample size. The resulting error will be reduced by a factor of . The same goes for any significance testing. Increasing the sample size will reduce the error in a similar manner.

Team experiment

Break into 3 teams

Team one will be using 2 dice

Team two will be using 4 dice

Team three will be using 6 dice

Each team will conduct 100 throws of their dice and record the average of each throw.

Plot a histogram of the resulting data.

Each team presents the results in a 10 min report out.

(

)