1. Process Capability Analysis
updated
Design for Manfucturability
Dr. Jerrell T. Stracener, SAE Fellow
SMU
ME 5350/7350

2. Measures of Process Capability
Statistical Analysis of Process Capability

3. Process Capability
Refers to the uniformity of the process.
Variability in the process is a measure of the
uniformity of the output.
- Instantaneous variability is the natural or
inherent variability at a specified time
- Variability over time

4. Process Capability
A critical performance measure that addresses
process results relative to process/product
specifications.
A capable process is one for which the process
outputs meet or exceed expectation.

5. Process Capability Measures or Indices
Process capability indices are used to measure the
process variability due to common causes present
in the process
The Cp index
Inherent or potential measure of capability
specification spread
process spread
The CpK index
Realized or actual measure of capability
Other indices
CpM, CpMK
Cp =

6. Measures of Process Capability
Customary to use the Six Sigma spread in the
distribution of the product quality characteristic

7. Key Points
The proportion of the process output that will fall
outside the natural tolerance limits.
Is 0.27% (or 2700 nonconforming parts per million)
if the distribution is normal
May differ considerably from 0.27% if the
distribution is not normal

9. Measure of Potential Process Capability, Cp
Cp measures potential or inherent capability of the
process, given that the process is stable
Cp is defined as
, for two-sided
specifications
and , for lower
specifications only
, for upper

10. Interpretation of Cp
is the percentage of the specification band used up
by the process

11. Measure of Potential Process Capability, CpK
CpK measures realized process capability relative to
action production, given a stable process
Cp is defined as

12. Interpretation of CpK
< 1, then conclude that the
process is stable
If CpK = 1, then conclude that the
process is marginally capable
> 1, then conclude that the
process is capable

13. Recommended Minimum Values of the Process
Capability Ratio
Two-sided One-sided
specifications specifications
Existing process
New processes
Safety, strength,
or critical parameter
existing process
new process

14. Process Fallout (in defective ppm)
PCR One-sided specs Two-sided specs

15. Estimation of Process Capability Ratios

16. Estimation of Cp
A point estimate of Cp is:
where

17. Estimation of Cp - example
If the specification limits are
LSL = and USL = and

18. Estimation of CP - example
and the process uses
of the specification band.

19. Example
To illustrate the use of the one sided process
capability ratios, suppose that the lower specification
limit on bursting strength is 200 psi. We will use
= 264 and S = 32 as estimates of  and ,
respectively, and the resulting estimate of the one
sided lower process capability ratio is

20. Example
The fraction of defective bottles produced by this
process is estimated by finding the area to the left
of Z = (LSL - )/ = ( )/32 = -2 under
the standard normal distribution. The estimated
fallout is about 2.28% defective, or about 22,800
nonconforming bottles per million. Note that if the
normal distribution were an inappropriate model
for strength, then this last calculation would have
to be performed using the appropriate probability
distribution. ^

21. Uses of Results from a Process Capability Analysis
1. Predicting how well the process will hold the
tolerances.
2. Assisting product developers/designers in
selecting or modifying a process.
3. Assisting in establishing an interval between
sampling for process monitoring.
4. Specifying performance requirements for new
equipment
5. Selecting between competing vendors.
6. Planning the sequence of production processes
when there is an interactive effect on processes or
7. Reducing the variability in a manufacturing
process.

22. Statistical Analysis of Process Capability

23. The Situation
In many situations, our knowledge is limited to the
information that can be obtained from data that has
been obtained or that will be obtained

24. The Problem
The challenge is to obtain the maximum information
from the data and to arrive at the most accurate
conclusions

25. Nature of Data
Most data are characterized by variation, as opposed
to deterministic, due to variation in
Processes and materials
Product/Manufacturing
Inspection & Measurement
Operation
Environment
etc

26. Need
Methods and techniques are needed for analysis of
data that account for
Variation in the data
Uncertainty in conclusion

27. Statistics
Statistics is the science of analyzing data and
drawing conclusions
Statistical methods and techniques that provide
tools for:
- experimental design
- analysis of data
- making inferences

33. Example
The number of defects per inspected PC-X based on a random sample of 15 from a days’ production is: 0, 2, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 1, 2, 1
(i.) Analyze these data and present your results.
(ii.) Estimate the probability that a randomly selected PC will have at least 3 defects.

34. Example
Twenty-five customers selected at random were asked to rate the overall satisfaction, a measure of quality, with the PC-X. Five factors were ranked by each selected customer. Each factor was assigned a rank between 1 and 10, with 10 indicating the highest level of customer satisfaction. The ratings were averaged over the five factors for each customer. The results are:
(i.) Analyze the survey data and present your results.
(ii.) Estimate the proportion of the customer population whose average satisfaction rating is at least 9.

35. Basic Concepts
Analysis of Location, or Central Tendency
Analysis of Variability
Analysis of Shape

36. Population vs. Sample
Population
the total of all possible values (measurement,
counts, etc.) of a particular characteristic for a
specific group of objects.
Sample
a part of a population selected according to some
rule or plan.
Why sample?
- Population does not exist
- Sampling and testing is destructive

37. Sampling
Characteristics that distinguish one type of sample
from another:
the manner in which the sample was obtained
the purpose for which the sample was obtained

38. Types of Samples
Simple Random Sample
The sample X1, X2, ... ,Xn is a random sample if
X1, X2, ... , Xn are independent identically
distributed random variables.
Remark: Each value in the population has an
equal and independent chance of being included
in the sample.

39. Analysis of Data
Data represents the entire population
Statistical analysis is primarily descriptive.
Data represents sample from population
Statistical analysis
- describes the sample
- provides information about the population

40. Analysis of Location or Central Tendency
Sample (Arithmetic) Mean
Sample Midrange
Sample Mode
Sample Median
Sample Percentiles

41. Sample Mean
Formula:
Remarks:
Most frequently used statistic
Easy to understand
May be misleading due to extreme values

42. Sample Mode
Definition:
Most frequently occurring value in the sample
Remarks:
A sample may have more than one mode
The mode may not be a central value
Not well understood, nor frequently used

43. Sample Median
Formula: , if n is odd & K = (n+1)/2
, if n is even & K = n/2
where the sample values X1, X2, ... , Xn
are arranged in numerical order
Remarks:
Not well understood, nor accepted
All sample data does not appear to be utilized
Not affected by extreme values

44. Analysis of Variability
Sample Range
Sample Variance
Sample Standard Deviation
Sample of Coefficient of Variation

45. Sample Range
Formula: R = Xmax - Xmin
where Xmax is the largest value in the sample
and Xmin is the smallest sample value
Remarks:
Easy to determine
Easily understood
Determined by extreme values
Does not use all sample data

46. Sample Variance & Standard Deviation
Sample Standard Deviation
s = (sample variance)1/2
Remarks
Most frequently used measure of variability
Not well understood

47. Sample Coefficient of Variation
Sample Variance
Remarks
Relative measure of variation
Used for comparing the variation in two samples of
data that are measured in two different units

48. Analysis of Shape
Skewness
Kurtosis

49. Estimate of Skewness
For a unimodal distribution, xr is an indicator of
distribution shape
< 1 , indicates skewed to the left
xr = 1 , indicates symmetric
> 1 , indicates skewed to the right

50. Estimation of Skewness
Estimate of skewness of a distribution from a
random sample
where
and

51. Comparison of Kurtosis

52. Presentation of Data

53. Time Series Graph or Run Chart
Stem-and-Leaf Plot
Digidot Plot
Box Plot
Frequency Distribution
Histogram and Relative Frequency

54. Time Series Graph or Run Chart
A plot of the data set x1, x2, …, xn in the order
in which the data were obtained
Used to detect trends or patterns in the data
over time

55. Stem-and-Leaf Plots A quick way to obtain an informative visual
representation of the set of data x1, x2, …, xn
for which each xi consists of at least two digits
Steps for constructing a stem-and-leaf display
1. Select one or more leading digits for the stem
values. The trailing digits become the leaves.
2. List possible stem values in a vertical column.
3. Record the leaf for every observation beside the
corresponding stem value.
4. Indicate the units for stems and leaves
someplace in the display
The stem and leaf display does not take the time
order of the observed data into account

56. Stem-and-Leaf Plot - Example
Here are test scores for 25 students:
69, 55, 80, 95, 94, 98, 51, 70, 93, 57, 62, 52, 52
58, 61, 51, 64, 67, 78, 68, 69, 68, 96, 73, 71
The first step is to place the numbers in order from
least to greatest:
51, 51, 52, 52, 55, 57, 58, 61, 62, 64, 67, 68, 68,
69, 69, 70, 71, 73, 78, 80, 93, 94, 95, 96, 98

57. Stem-and-Leaf Plot - Example
Now create the graph:
Test Scores
8 0
The numbers on the left side of the vertical line are
the stems. The numbers on the right side are the
leaves. In this graph, the stems are the tens digits
and the leaves are the unit digits. In this case, 9|3
represents a score of 93.

58. Digidot Plot
A combination of the time series graph with the
stem and leaf display

59. Box Plot
A pictorial summary used to describe the
most prominent statistical features of the data
set, x1, x2, …, xn, including its:
- Center or location
- Spread or variability
- Extent and nature of any deviation from
symmetry
- Identification of ‘outliers’

60. Box Plot
Shows only certain statistics rather than all the
data, namely
- median
- quartiles
- smallest and greatest values in the
distribution
Immediate visuals of a box plot are the center,
the spread, and the overall range of distribution

61. Box Plot
Given the following random sample of size 25:
38, 10, 60, 90, 88, 96, 1, 41, 86, 14, 25, 5, 16,
22, 29, 34, 55, 36, 37, 36, 91, 47, 43, 30, 98
Arranged in order from least to greatest:
1, 5, 10, 14, 16, 22, 25, 29, 30, 34, 36, 36, 37,
38, 41, 43, 47, 55, 60, 86, 88, 90, 91, 96, 98

62. Box Plot
First, find the median, the value exactly in the
middle of an ordered set of numbers.
The median is 37
Next, we consider only the values to the left of
the median:
1, 5, 10, 14, 16, 22, 25, 29, 30, 34, 36, 36
We now find the median of this set of numbers.
The median for this group is ( )/2 = 23.5,
which is the lower quartile.

63. Box Plot
Now consider the values to the right of the
median.
38, 41, 43, 47, 55, 60, 86, 88, 90, 91, 96, 98
The median for this set is ( )/2 = 73, which
is the upper quartile.
We are now ready to find the interquartile range
(IQR), which is the difference between the upper
and lower quartiles, = 49.5
49.5 is the interquartile range

64. The interquartile range is 49.5
Box Plot 10 20 30 40 50 60 70 80 90
100
The lower quartile 23.5
The median is 37
The upper quartile 73
The interquartile range is 49.5
lower
extreme
upper
quartile
median

65. Histogram
A graph of the observed frequencies in the data
set, x1, x2, …, xn versus data magnitude to
visually indicate its statistical properties, including
- shape
- location or central tendency
- scatter or variability

66. Guidelines for Constructing Histograms
If the data x1, x2, …, xn are from a discrete
random variable with possible values y1, y2, …, yn
count the number of occurrences of each value
of y and associate the frequency fi with yi,
for i = 1, …, k

67. Guidelines for Constructing Histograms
If the data x1, x2, …, xn are from a continuous
random variable
- select the number of intervals or cells, r,
to be a number between 3 and 20, as an
initial value use r = (n)1/2, where n is the
number of observations
- establish r intervals of equal width, starting
just below the smallest value of x
- count the number of values of x within
each interval to obtain the frequency
associated with each interval
- construct graph by plotting (fi, i) for
i = 1, 2, …, k

68. Statistical Process Control- Histograms
Possible answers for a Cliff-like histogram
Hiding data that should be outside the specification
Supplier is screening the product before shipment
Lower specification is a physical limit like zero
thickness, but this is not normally the case
lower spec
upper spec

69. Statistical Process Control- Histograms
Possible answers for a Bimodal histogram
Two primary sources of process variation
The process is stable, but it has experienced
a large shift during the time the data were collected
lower spec
upper spec

70. Statistical Process Control - Histograms
Possible answers for a Comb-like histogram
Insufficient data collected
Too many classes displayed
Process is unstable
Process is stable but is multimodal
lower spec
upper spec

71. Statistical Process Control - Histograms
Possible answers for a Skewed histogram
May be the natural result of the process
For a machined part, the equipment may be losing
tolerance or tools may be wearing out
The process is shifting slowly to the side with the
long tail
lower spec
upper spec

72. Statistical Process Control - Histograms
By including specification limits on a histogram, the
amount of data that falls outside of the specification
limits can be easily seen
specification
frequency
lower spec
upper spec

73. Normal Distribution - Estimation of m &s
X1, X2, …, Xn is a random sample of size n from
n(, )
Point Estimate of 
Point Estimate of 

74. Random sample of size n, X1, X2, ... , Xn from Ln (, )
Estimation of Lognormal Distribution Estimation of m & s
Random sample of size n, X1, X2, ... , Xn from
Ln (, )
Let Yi = lnXi for i = 1, 2, ..., n
Treat Y1, Y2, ... , Yn as a random sample from
N(, )
Estimate  and  using the Normal Distribution
Methods

75. Random sample of size n, T1, T2, …, Tn, from W(, ) Point estimates
Estimation of Weibull Distribution - Estimation of b & q
Random sample of size n, T1, T2, …, Tn, from
W(, )
Point estimates
 is the solution of g() = 0
where