1. Process Capability Studies
Comparison of Process Output to Product Requirements
By Lynn Taylor OISM 470W - Section T, Th 4:15-5:30
Lynn Taylor – OISM 470W Section 002

2. Process Capability Studies 3 Steps
Gauge Capability
Process Characteristics
Stability
Normality
Control
Process Capability Analysis
Comparison of process output(s) to requirements.
Step 1 – Gauge Capability
Before studying the capability of a process by using measurement data, we must first be sure that our test methods, equipment, operators, and conditions are accurate, robust, and capable of measuring to much greater precision than our tolerances require.
Step 2 – Process Characteristics
The second step in the study is to examine our process measures over several time periods and use these data to examine the output for special causes of variation (outliers) which should be identified and eliminated from the process before continuing with the study. Several experiments may be required before eliminating all assignable causes.
Step 3 – Process Capability Analysis
Comparison of process data to product specifications may indicate the need to adjust the process, innovate, or adjust the specification. The statistics generated by a process capability analysis are the process capability index, Cp, the upper process capability index, CpU, and the lower process capability index, CpL. The lesser of the two one-sided indices, CpL and CpU, is referred to as Cpk. These indices are calculated as follows:
Cp=(USL-LSL)/6σ where: USL is the upper specification limit
CpU=(USL-Xbar)/3σ LSL is the lower specification limit
CpL=(Xbar-LSL)/3σ σ is the process standard deviation (of individuals)
Cpk=the lesser of CpU and CpL Xbar is the process average for the period studied
Typically, we like to see a Cpk of at least 1.33, which indicates that the control limit of our test result (for individual values) that is nearest to its tolerance limit is at least 1σ from that limit. This allows us a “safety cushion” should a new special cause of variation shift the location of our distribution or widen it.

3. Process Capability Studies Definitions
Process – Work performed on a set of inputs to produce a result.
Capability – The predictable result(s) of a process allowed to operate without outside interference.
Process Capability Study – Comparison of process output(s) to product requirements.
Process
A process is the work performed with a set of inputs – raw materials, machines, and employees, that adds value and provides a product or service for the customer (internal or external).
Capability
Process capability is the ability of a process to produce products or services in a state of statistical control (predictable) for a period of time under a given set of conditions. Outside influences (special causes of variation) must be eliminated from the process in order to obtain this level of statistical control.
Process Capability Study
After determining that our gauge(s) are capable of measuring to the level of precision necessary and that our process is operating in a state of statistical control, we can perform the process capability analysis. This final step is the comparison of our process spread, as defined by the + 3 spread of a normal distribution (for measurement data) or the location and spread of the Poisson distribution (λ for attributes) to our product or service tolerances. We will only look at variable data, not attributes, in this seminar.

4. Process Capability Studies Uses In Our Plant
Customer 1 < 407, Cpk > 1.25
Customer 1 > 407, Cpk > 1.50
Customer 2, Cpk > 1.33
With new clients, determine if tolerances are reasonable, if our processes are capable and robust, and if improvements are needed.
Current Uses
We have customer requirements for Cpk from two customers. Customer 1’s requirements are based upon the last 32 observations in a production run while the customer 2 indices are reported monthly, on the past month’s manufacturing activity.
Additional Uses
Beyond current customer requirements, we can use capability studies to:
Determine if our measurement systems are capable of meeting our needs for new parts.
Uncover process variables affecting new parts which need to be controlled or eliminated.
Point out whether new or different processing equipment or processing steps are needed in order to make the best possible part.
Evaluate initial customer tolerances to see if they should be widened, shifted, or tightened, based on the capability of their tooling.

5. Process Capability Studies Step 1 – Gauge Capability
Six Measures
Gauge Accuracy
Are the readings correct?
Gauge Linearity
Are the readings correct over the measurement range of the instrument?
Gauge Stability
Does the gauge return similar values over time?
Gauge Accuracy
Accuracy is assured by gauge calibration. During calibration, the gauge in question measures a known standard, traceable to a master standard maintained by the National Institute of Standards and Technology (NIST) in Gaithersburg, MD, or another recognized standards body. The gauge must return a value within a previously specified range (dependent on the gauge design and intended use, and required precision for the measurement). If it does not, the gauge is either adjusted, repaired, or replaced.
Gauge Linearity
Gauge linearity references the accuracy and precision we can expect of the gauge at different levels of output. Gauge calibration includes taking several measurements, across the range of outputs that we expect to use the gauge in order to quantify accuracy at each level. We could do a Gauge Repeatability and Reproducibility (GRR) study at each level to determine the expected spread (range) of results from the gauge at each level.
Gauge Stability
Gauge stability involves measuring the GRR over time to determine drift and changes in the system. To avoid drift, we calibrate gauges at a frequency that has proven to be less than that with which the gauge has shown a significant shift in the past.

6. Process Capability Studies Step 1 – Gauge Capability
Six Measures
Gauge Repeatability
Does one operator return a similar result time after time (gauge variability)?
Gauge Reproducibility
Do different operators return similar results?
Gauge Repeatability and Reproducibility
Total variability of a test result (operator & gauge).
Gauge Repeatability
Gauge repeatability is the maximum permissible difference, due to test error, between two test results obtained by the same operator, using the same gauge, on the same test specimen (gauge variability).
Gauge Reproducibility
Gauge reproducibility is the maximum permissible difference, due to test error, between two test results obtained by two different operators, using the same gauge, on the same test specimen (operator variability).
Gauge Repeatability and Reproducibility
Gauge Repeatability and Reproducibility (GRR) is the total variability of a test result (includes operator and gauge variability). Some sources, such as ASTM, define this statistic as simply reproducibility, while others, such as General Motors (GM), use the GRR designation. It is important to know which is being referenced, we use the GM designations.
Since we have performed GRR’s on our measuring machine for the parts we will look at later, and it is capable, we will not address gauge capability in this training session.

7. Process Capability Studies Step 2 – Process Characteristics
Analyze Process Data
Normality – Evidenced by the descriptive statistics skewness and kurtosis.
Stability – The absence of shifts or trends over time.
Control – Control chart analysis passing tests for special causes.
Normality
The measures of normality we will apply to our distribution are skewness and kurtosis. Skewness is a measure of symmetry, it quantifies if the distribution is significantly wider on one side than the other. A perfectly normal distribution has a skewness of 0. The maximum skewness for a normal distribution is +7.45/N where N is the number of observations. Kurtosis is the measure of steepness or peakedness of the distribution. A normal distribution has a kurtosis of 3, higher numbers indicate a steeper slope, smaller numbers indicate a flatter curve. Maximum/minimum kurtosis for a normal distribution is 3 + (14.9/N), where N is the number of observations. A distribution that is flatter than normal is called platykurtic, one that is steeper than normal is called leptokurtic.
Stability
Control chart analysis will be used to look for trends or shifts in the data over time. These are evidenced as a constantly rising or falling graph, or by sudden shifts upward and/or downward on the chart.
Control
Visual inspection of the Xbar and R control chart allows us to check for special causes of variation. The laminated cards you received list five of these tests for special causes of variation but there are many more. The main thing to remember is that, in nature, variation is RANDOM. Any chart that appears to have some special pattern is probably being acted upon by some special cause(s) of variation and should be investigated.
The following slides illustrate the concepts of normality, stability, and control.

8. Process Capability Studies Tests for Normality
There are four statistical measures, called “moments”, which describe all distributions. We should all be familiar with the first two, the others may be new to you:
Mean –The mean is the location of the distribution, the midpoint if the distribution is normal. There are two other measures of location which are used for other purposes, these are the median, which is the middlemost value in the distribution, and the mode, which is the most frequently occurring value. If the distribution is normal, these three statistics will be the same, if not, they will be different.
Standard Deviation – The standard deviation is a measure of the spread (width) of the distribution. The top left distribution on the slide above shows the shape of the normal curve and the expected percentage of observations we would expect to see within each 1σ cell of the distribution.
Skewness – Skewness is a measure of symmetry of the distribution, a perfectly normal distribution has a skewness of 0, the maximum skewness we can still call normal is equal to 7.49/N, where N is the number of observations. The distribution in the lower left corner of the slide shows a distribution that is skewed to the right, it has a positive skewness. If the pattern were reversed, the skewness value would be a negative number.
Kurtosis – Kurtosis is the measure of steepness of the distribution’s curve. The kurtosis of a normal curve is 3. The distribution in the top right corner of the slide is flatter than normal (platykurtic) and would have a kurtosis less than 3. The distribution in the lower right corner is steeper than normal (leptokurtic) and would have a kurtosis greater than 3. The maximum/minimum values for a normal kurtosis are given by 3 + (14.9/N), where N is the number of observations.
We will use the skewness and kurtosis measures later to determine if our distribution is normal and we will use the average and standard deviation to calculate our process capability indices.

9. Process Capability Studies Tests for Stability and Control
Pattern 1, above, depicts a control chart of a measurement of a process operating in a state of statistical control. Measurements take up most of the space between the control limits, with the majority of the values falling in the center third of the chart.
Charts 2 and 3 both show significant shifts which indicate an unneeded adjustment has been made to our process or some other major change has occurred. Chart 3 seems to have been corrected, knowingly or not, back to its original level.
Graph 4 shows a steady slide downward. This may be indicative of a slipping adjustment (mold water warming up) or a worn fixture.
Traces 5 and 6 show a cyclic pattern. Number 5 may be the result of one team vs another over time. Number 6 may be an operator 1 vs operator 2 (or machine 1 vs machine 2) situation.
Most people would say there’s nothing wrong with chart 7. The problem is, it’s TOO GOOD. Remember, only 68% of charted values should lie between + 1σ from the average, all of these values lie in this range. Something’s up here.
Chart 8 is what we react to most readily, a measurement outside control limits. Oftentimes, this is just a measurement error which can be rechecked and corrected.
Patterns like charts 2 through 8 are all cause for investigation and indicate a lack of stability and control of the process.

10. Process Capability Studies Comparing Distributions to Specifications
The curves above illustrate how different populations of measurements will be represented by the process capability indices Cp, CpL, and CpU.
The three distributions on the left are all perfectly centered within their tolerance window. In instances like these (rare) the CpL and CpU values will exactly equal the Cp value. The top curve lies well within the specification range and gives us a Cp of 1.67, quite capable. The middle curve JUST fits within the specification range, returning a Cp value of Any shift in the process will likely cause some of this product to fall outside tolerances. The bottom distribution is too wide to consistently make product within specifications. The Cp index of 0.71 tells us that we will always have to do final inspection to cull out the rejects made by this process.
The distributions on the top right display the Cp, CpU, and CpL values for three distributions that are too wide to consistently meet specifications and all three have the same spread, hence the same Cp index. The first is centered in the tolerance range but is making product both too large and too small, both CpL and CpU are equal, but less than The second distribution has shifted downward, so the CpU is 1.00, but the CpL has dropped to The third distribution has shifted up and is making all product above the lower specification limit, but is making a significant proportion above the upper spec. In all instances, Cpk is the lesser of CpL and CpU.
The three distributions on the bottom right could all be capable, as evidenced by Cp indices of 1.0 or greater, except that their location has shifted, causing all to produce parts outside of tolerances. Adjustment of these processes is needed to bring their product within tolerances. Look at the respective Cp, CpU, and CpL indices to see how the location affects the capability of the process.

11. Process Capability Studies Real World Example – Part 504
Real World Example – Part 504 Tests for Normality
Let’s take a look at a real example from the lab. The last 100 observations of the 9.56 dimension for 504 parts were analyzed to determine normality. Looking at the histogram, we can see that the distribution is skewed to the right and its location (mean) is also to the high side of the tolerance range. The QC-Calc program has pronounced the skewness of our data to be non-normal. Using our formula for maximum skewness for a sample size of 100, we get 7.49/100 = Since that value is greater than our distribution’s skewness of 0.49, we’ll consider our distribution to be normal enough to use for capability analysis. The program indicates that the kurtosis is normal with a value of Using our formula 3 – (14.9/ 100 ) gives us a minimum acceptable kurtosis of 1.51, our curve is much steeper than that so we are ok with this statistic as well.
Next, we’ll look at how well controlled our process is.

12. Process Capability Studies Part 504 Continued
Real World Example Part 504 – Tests for Control
The first thing we check on an Xbar and R chart is the range chart, to be sure it is in-control. We see all the ranges above are in-control so we can focus attention on the Xbar chart. Although the program automatically places the specification limits on the chart, we can ignore them at this point since the specifications only apply to individual values, not to subgroup averages. Since we do measurements in subgroups of four, that is how the data are plotted. All the plotted points fall within the control limits and take up a reasonable amount of the spread between the control limits. The pattern appears to be fairly random and normal-looking. One thing to note, the lower control limit is right at the nominal, so all of our parts are slightly above the target for this cavity. The nominal for this dimension is 9.520mm while our overall average measurement is 0.008mm above this target. Since everything looks in order with the control chart, let’s move on to see the process capability indices. Let’s calculate them before moving to the next slide and use QC-Calc to check our work, using the mean given above (Xbarbar= ), the standard deviation from the histogram on the previous slide ( ), and the upper and lower specifications given above. Using these statistics we get:
Cp = (USL – LSL)/(6σ) CpL = (Xbarbar – LSL)/(3σ)
Cp = (9.56 – 9.48)/(6 * ) CpL = ( – 9.48)/(3 * )
Cp = 0.080/ CpL = /
Cp = CpL = 2.64
CpU = (USL – Xbarbar)/(3σ) Cpk = the lesser of CpL and CpU
CpU = (9.56 – )/(3 * ) Cpk = 1.73
CpU = /
CpU = 1.73
Now let’s see what numbers QC Calc got.

13. Process Capability Studies Part 504 Continued
Comparing the values we calculated for Cp, CpU, CpL, and Cpk to those above shows we have found the answer to our question. Even though the part averages 0.008mm above the nominal, our process is still capable of meeting customer expectations for this dimension from this part.
Next, let’s do a sample problem to check our understanding of the concept.

14. Process Capability Studies Part 613 - An Exercise
The 100 values listed above are the dimension from part 613. The first thing we’ll do is construct a histogram and calculate the mean and standard deviation for the individual data. Since the calculations for skewness and kurtosis are not on our calculators, and they are rather involved, we’ll use the computer to calculate them for us. To do your histogram, use the following lower bounds for each cell:
0.1077, , , , , , , , and

15. Process Capability Studies Part 613 – An Exercise
Your histogram (hopefully) should look similar to the one above. The computer also calculated the descriptive statistics for us, but let’s just take a look at the distribution first. The curve looks fairly symmetrical, as evidenced by the low skewness number, and the slope looks similar to the normal curve drawings we have seen so far, evidenced by the kurtosis of slightly more than 3. The one fly in the ointment is the average value of , we’re off our target by almost 0.002, not good when our tolerance is just Let’s work through this and see how our numbers turn out. Next, we’ll construct our Xbar and R charts to determine stability and control.

16. Process Capability Studies Part 613 – An Exercise
First, we’ll take a look at the range chart. All the ranges fall within the upper control limit and all ranges are well under 0.001, indicating we are pretty consistent from part-to-part.
Now let’s look at the X-bar chart. All points are between the control limits and we are not stratified around the center of the distribution. What does jump out is that there are 10 points in a row (observations 11-20) which fall below the average. Investigation for possible reasons, checking process, material changes, and recycle content, yielded nothing had changed here. The one factor I couldn’t check was age of the parts when measured. If the parts cooled longer than normal before measurement, they could have measured slightly smaller. Since no special cause could be found for the run below the centerline, we must use these data as indicative of our process.
We’ll move on to our next slide in a minute, first let’s do the calculations to get our process capability indices.
Cp= (USL-LSL)/6σ where USL = 0.115, LSL = 0.105, and σ = (from the previous slide)
CpL= (Xbar-LSL)/3σ where Xbar is
CpU= (USL-Xbar)/3σ
Cpk= the lesser of CpU and CpL
Got it? Let’s check your answers……………………………………

17. Process Capability Studies Part 613 – An Exercise
And the answer is:
As you can see, our process is quite capable, even though we are off-center by almost This is not much incentive to invest the thousands of dollars the customer would have to invest in order to modify the tooling to center the hole size within the tolerance – WE’RE TOO GOOD! Our Cpk is 6.4, it’s a very slim possibility we’ll ever make a bad part. Let’s recap what we’ve gone over.

18. Process Capability Studies Summation
Three Steps:
Gauge Capability
Process Characteristics
Process Capability Analysis
We can only do the process capability analysis after we have checked the accuracy and variability of our measurement systems. This includes calibration and GRR analysis. We must repair or replace the gauge or retrain operators if either of these activities indicates an unacceptable result.
We must look at our measurement data graphically, using histograms and control charts, as well as using descriptive statistics (mean, standard deviation, skewness, and kurtosis) to be sure our process producing the part being measured is stable, normal, and in control. If our data fail any of these tests, we must investigate our process and eliminate the special cause(s) of variation that caused the abnormal pattern(s).
The final step (and, by far, the easiest) is to calculate the capability indices from the available data to quantify the capability of our process.

19. Process Capability Studies Bibliography
Introduction to Statistical Analysis – sixth ed – Dixon and Massey
Introductory Statistics – Weiss and Hassett, 1982
AT&T Statistical Quality Control Handbook – 11th Printing – 1985
Experimental Statistics – Mary G. Natrella – 1966 (NBS Handbook 91)
General Motors Statistical Quality Control Handbook – 1982
LTV Steel Integrated Process Control Handbook – Norman Bresky and Joan Harley – 1986
North American Refractories Co. SPC Handbook – 1988
1994 Annual ASTM Book of Standards- ASTM E