SPC tools and techniques– part 2 & Fundamentals of Statistics—Part 1
Zatan_07
Lecture8.pdf
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Lecture 8: Fundamentals of Statistics—Part 1
Chapter 5
Outline
Statistics, type of data, data description
Frequency distribution and histogram
Measures of central tendency
Measures of dispersion
Population, sample,
Normal curve
Computer program
Types of Data
Variable
Quality characteristics that are measurable, continuous.
Example: length, weight
Attribute
Classified either conforming or nonconforming (yes/no, accept/reject)
Go/NoGo gauge
A Go/no go refers to an inspection tool used to check a workpiece against its allowed tolerances. Its name derives from its use: the gauge itself has two tests; the check involves the workpiece's having to ‘pass’ one test (Go) and 'fail' the other (No Go).
Go/NoGo gauge
Lower gauge is a plain plug gauge of 12.60mm for 'Go' and 12.90mm as 'NoGo‘ For checking tolerance between 12.60-12.90.
For a washer with dia= 12.75, pass one and fail the other and a washer with dia= 12.5, fail at both end
Describe Data
Summarizing data:
Graphical (describe picture)
Plot or picture of frequency distribution
Analytical (math calculations)
Measures of central tendency (mean, mode, median)
Measures of dispersion (range, std. deviation)
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Descriptive statistics
Mean = average()
Median = median()
Mode = Mode()
Std. Dev. = STDEV()
Variance = var()
Count = count()
Max = max()
Min = min()
Range = max()-min()
Skewness = skew()
Kurtosis = kurt()
Sum =sum()
Student scores
82.64, 84.47, 77.12, 79.08, 95.21, 81.84, 94.66, 80.50, 84.96, 89.18, 87.76
Mean=85.22
Median=84.47
Max= 95.21
Min=77.12
Describe Data
0
1
2
3
4
5
6
7
<75 <80 <85 <90 <95 <100
Frequency
Frequency
Frequency Distribution
Ungrouped data: a listing of the individual observed values
Grouped data: a lumping together (subgroups) of the observed values
Frequency Distribution– ungrouped data
A table that includes the number of daily billing errors of a larger organization
Example— number of daily billing errors
Organize the data (array)
Tabulate the frequency of each value
Different types of frequency distributions
Histogram: consists of a set of rectangles that represent the frequency in each
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Different types of frequency distributions
Relative frequency distribution: proportion or fraction of the total
Different types of frequency distributions
Cumulative frequency
Different types of frequency distributions
Relative cumulative frequency
Summarization of this billing example
This example shows how to organize and sort the data by frequency, relative frequency, cumulative frequency, and relative cumulative frequency.
We have practiced this tasks in Pareto chart
Students should be able to chart these frequency data by Excel
Frequency Distribution– grouped data
Sometimes the collected data is a large volume of data set. If we still use the previous method, the frequency distribution result may not give a good explanation.
Frequency Distribution– example of grouped data
Example: steel shaft weight (kilograms)
Simplify data by coded value: the weights are coded from 2.500Kg, a weight with a value of 31 is equivalent to 2.531Kg (2.500+0.031)
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Steel shaft weight
Total: 110
Highest: 75
Lowest: 31
Frequency Distribution– grouped data
Collect data and construct a tally sheet
Determine the range
Determine the cell (subgroup) interval
Determine the cell midpoints
Determine the cell boundaries
Post the cell frequency
1. Data tabulation: from lowest to highest, a tally sheet
A large number of categories
2. Determine the range
Where R=range
Xh= highest data
Xl = lowest data
044.0531.2575.2 lh
XXR
How many cells (categories) should be formed– grouping
Number of cells is based on judgment
General rule of thumb is the number of cells should be between 5 and 20
5 to 9 when observations are < 100
8 to 17 observations are between 100 & 500
15 to 20 when observations are > 500
Three issues in creating the cells
Equal width of cell intervals
Cell midpoints
Cell boundaries
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3. Determine the cell interval width
Cell interval is the distance between adjacent cell midpoints. Odd number is preferred because of easy calculation
0057.0 )110log(322.31
044.0
log322.31
n
R i
3. Determine the cell interval length
All trial-and-error method, here h is the total number of intervals needed
Assume i=0.003, then
Assume i=0.005, then
Assume i=0.007, then
9 005.0
044.0
i
R h
15 003.0
044.0
i
R h
6 007.0
044.0
i
R h
4.Determine the cell boundaries and midpoints
When constructing a histogram, it’s important to remember two things:
Histogram must contain all of the data
One particular value cannot fit into two different cells, which means cells cannot overlap
4.Determine the cell boundaries and midpoints
1st method-- the simplest technique is to choose the lowest value measured as first midpoint, and so on…
2nd method-- lowest value will be the lower boundary for first cell (then, the first midpoint would be the lowest value plus half of interval), and so on…
2
i XMP
ll
4.Determine the cell boundaries and midpoints— by 1st method
Lowest value as the first midpoint → so
first Midpoint =2.531
Since width=0.005, so first cell boundaries:
Lower-B=2.531-0.005/2
= 2.5285
Upper-B=2.531+0.005/2
=2.5335
Midpoint Cell boundaries
2.531 2.5285 - 2.5335
2.536 2.5335 - 2.5385
2.541 2.5385 – 2.5435
2.546 2.5435 – 2.5485
2.551 2.5485 - 2.5535
2.556 2.5535 - 2.5585
2.561 2.5585 – 2.5635
2.566 2.5635 – 2.5685
2.571 2.5685 – 2.5735
2.576 2.5735 – 2.5785
4.Determine the cell boundaries and midpoints— by 2st method
Lowest value as the lower boundary of the first cell → so Lower-B=2.531,
Adjust a little bit, then
Lower-B=2.530
First Midpoint =2.530+0.005/2 = 2.5325
Upper-B=2.530+0.005
=2.535
Midpoint Cell boundaries
2.5325 2.530 - 2.535
2.5375 2.535 - 2.540
2.5425 2.540 - 2.545
2.5475 2.545 – 2.550
2.5525 2.550 - 2.555
2.5575 2.555 - 2.560
2.5625 2.560 – 2.565
2.5675 2.565– 2.570
2.5725 2.570 – 2.575
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4.Determine the cell boundaries and midpoints
Other midpoints: 2.533+0.005=2.538
2.538+0.005=2.543…..
533.2 2
005.0 531.2
2
i XMP
ll
4.Determine the cell boundaries and midpoints
Boundaries are established so there is no question as to the location of an observation.
2.530 2.535 2.540 2.575 2.545 2.550 2.555 2.560 2.565 2.570
6. Post the cell frequency Excel Histogram function
Histogram command
Input range — entire data
Bin range — using upper boundary to separate the cells
Given these two parameters, Excel will generate frequency column
Be careful: you must have “Data Analysis ToolPak” loaded
Characteristics of Graphs
Symmetry or lack of symmetry of the data (Distribution around the central value, Skewness)
Number of peaks
Peakedness of the data
Platykurtic (flat shape)
Leptokurtic (clear peak)
Characteristics of frequency distributions
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Analysis of Histogram
Many cases are expected to have a normal pattern (bell)
Graphical representation can help figure out problems if there is any in the process.
If not normally distributed, then something in the process may be out of control or the data follows some other basic pattern
Differences due to location, spread, and shape
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Histogram Computing function with Excel
Define the data range
Find out mean, median, mode, standard deviation
Construct histogram
Name “data” or other names Use functions to describe data Define midpoint and U-boundary Tool— Data Analysis—Histogram
Input range — data Bin range— U-boundary Output range—
Chart data— midpoint and frequency columns One useful website about histogram with
Excel
Procedures of using Excel to make histogram
How many cells
or categories
Boundaries of
cells or categories
Excel histogram
function (data, Bin) Chart
function
R=max(data)-min(data)
i=R/(1+3.322log(n)) # of cells = R/i
The very left cell,
L-bound
U-bound= L-bound + i
U-bound (n+1) = U-
bound (n) +i
Cell
midpoint Midpoint=(L-bound + U-bound)/2
Midpoint=L-bound + i/2
Assignment today/this week
InClass 6: Histogram function in Excel
Homework 2
