Cpk A Guide to Using a Process Capability Index Cpk = Min [ USL _ X, X _ LSL ] 3s 3s Specification Width Process Spread LSL X USL
The following information is provided by the Technology Issues Committee of AMT The Association For Manufacturing Technology to assist in the use and understanding of the Process Capability Index Cpk. Published by: AMT The Association For Manufacturing Technology 7901 Westpark Drive, McLean, VA 22102 Printed in the United States of America Copyright 2002 AMT The Association For Manufacturing Technology all rights reserved. Permission to photocopy this material is granted provided credit is given to AMT. For ordering information, see the Publication section of the AMT Website at www.amtonline.org or contact the AMT Information Resource Center (IRC) at 703-827-5220 or 703-893-2900.
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A Guide to Using C pk a Process Capability Index Introduction The purpose of this guide is twofold. The first is to provide information on the process capability index Cpk. The second is to list various actions that can be taken or parameters checked in order to reduce process variation. The idea of comparing the specification of a part parameter to the measured variation or distribution of the process producing the parameter has been with us for many years. It has only been in recent years that the comparison has been given a formal name and a means of calculation. All authors and analysts writing on Cpk hasten to point out that the index is a statistic based on measurements and, like all such statistics, has an associated degree of uncertainty. However, most practitioners consider Cpk to be a fixed number without regard to the nature of the data that produced it. We will point out the uncertainty involved in any statement of Cpk. This guide assumes the reader has knowledge of control charts and methods for calculating standard deviations. A good reference is the NIST/SEMATECH Handbook of Statistical Methods. The complete Handbook is on the Internet and may be accessed at www.itl.nist.gov/div898/handbook. What is C pk? Cpk is a Process Capability Index. The term index is used because the value is a comparison or ratio. It is the ratio of the 1
workpiece specification or tolerance (allowed variation) compared to the process variation (produced variation) expressed in terms of ± 3 standard deviations. When standard deviation is used in a calculation, the assumption is that the underlying measurements form a normal distribution. Therefore, in the case of calculating Cpk, all known assignable causes for variation in the process should be minimized before measurements are taken that will be used in the final calculation. In other words, the process should be stable and in statistical control. Some processes may use a positive stop or an in-process gage to produce part size. In those cases, the size distribution may not be normal, and the calculations described here will not be valid. Other sources should be consulted on how to deal with skewed distributions. In other cases, the specification is not bimodal nor is it given as a range. Examples might be hardness at least or surface finish not to exceed. In those situations a Cpk cannot be calculated since the part specification is not stated as a range. Of course, the standard deviation for the process output can still be calculated, and an estimate made about the probability of staying within the specification. But this is not a comparison such as Cpk. The importance of sample size in acquiring data cannot be over emphasized. As we shall see, the calculated value of Cpk depends on what is technically termed an estimate of the standard deviation. The larger the sample size, the more accurate is the estimate. 2
Calculating C pk Once data on the process has been gathered and analyzed, and the standard deviation calculated, a comparison to the product s specification can be made. This simple comparison yields the process potential Cp. In some cases, the mean of the process is at the center of the product s specification limit as shown in Figure 1. The term Cp assumes centering and should be equal to or greater than 1. Specification ion Width USL LSL Cp = = Process Spread 6s USL is the upper specification limit LSL is the lower specification limit X is the process mean Note: The symbol σ is used for standard deviation when very large samples are used that accurately represent the total population. In most cases, it is not feasible to use large samples, and the resultant standard deviation is represented by s. 3
However, in most cases, the process will not be centered on the specification as shown in Figure 2. The actual process capability Cpk then becomes Cpk = X Nearest Specification ion Limit 3s This is usually stated as USL X X LSL Cpk = Min [, ] 3s 3s FIG. 2 In Figure 2, the nearest specification limit is USL. An inspection of Figure 2 will show that the first step in increasing Cpk should be to take action to align the center of the process spread with the center of the specification spread. This assumes that the two spreads are close to equal, or the process spread is actually less than the specification spread. 4
Of course, if the process spread greatly exceeds the specification spread, steps must be taken to reduce the process spread. In Figure 2, Cpk would be about 0.5. However, if the process spread were aligned with the specification, Cpk would be about 1.0. Putting C pk in Perspective For the sake of simplicity, let s assume that the process is centered on the product specification. How many defective parts per million (parts out of tolerance) would we expect for different values of Cpk? Table 1 lists some values: Table 1: Expected number of defective parts for values of Cpk Cpk Parts per million defective 1.00 2,700.0 1.10 967.0 1.20 318.0 1.30 96.0 1.40 26.0 1.50 6.8 1.60 1.6 1.70 0.34 1.80 0.06 2.00 0.0018 It should be noted that a Cpk of 2 equates to roughly two parts per billion defective! Such a number highlights the significance of sample size and the related issue of uncertainty associated with the actual value of Cpk. 5
Suppose we would like to start with a 90% confidence level that a calculated value of Cpk based on measured data is equal to or greater than a specified value. What value would we have to see based on various sample sizes? Table 2 provides some examples: Table 2: Required Test Cpk Values for 90% Confidence in Specified Value Specified Value for Cpk Sample Size 1.00 1.30 1.50 1.70 2.00 200 1.08 1.40 1.61 1.82 2.14 100 1.11 1.44 1.66 1.88 2.21 50 1.17 1.51 1.74 1.97 2.31 30 1.24 1.60 1.84 2.07 2.45 10 1.50 1.93 2.22 2.52 2.95 Most experts agree that the sample size should be at least 30. For derivation of how to calculate the values in Tables 1 and 2 above, see the referenced NIST/SEMATECH Handbook (noted on page 1), section 7.1.4. Things to Remember About C pk Cpk is used to provide some expectation about the future capability of a process. However, the number calculated is based on a snapshot of the process at only one point in time. The calculated Cpk is only an estimate of how the process might be expected to perform. The confidence level we can assign to the calculated value is a function of sample size. 6
We should not lose sight of the fact that establishing process capability gives us a benchmark for improvement. Continuous improvement is the ultimate goal of making the measurements. Factors to Consider in Improving C pk Measurement The key element in establishing Cpk with a customer is reaching agreement on the measurement method and gauges to be used. The condition of the measurement equipment, and gauge reproducibility and repeatability (R&R) should be stated. In fact, for tight tolerances, the conditions used to determine R&R should be stated such as the number of appraisers and the number of repeat measurements. In order to be able to analyze data for events that happen during a test run, make sure that measurements are recorded chronologically. See Section 2.4 in the NIST/SEMATECH referenced Handbook for a complete discussion on gauge R&R. Machine Thermal deformation is one of the greatest contributors to change in the output of a machine tool. All elements responding to temperature change should be understood and monitored. Machine accuracy and repeatability should be determined using statistical techniques. Factors such as alignment, spindle runout and balance, and dynamic stability should be accessed with respect to the contribution to desired workpiece parameters. Machine maintenance is useful to restore parameters that have deteriorated and are contributing to variations. Company procedures should be established for maintaining machine calibration. 7
Tooling Changes in tool condition are a common source of shift in workpiece size or surface finish. These changes can best be analyzed from a histogram of data taken chronologically. Changes are not limited to tool wear, but may also be created by dirt on the toolholder, a balance condition, or repeatability when changing inserts. Workholding The ability of the workholding device to position each part consistently is critical to maintaining uniform output. Tests should be made to determine the repeatability of workholding devices. The rigidity of the workholding device in relation to the rigidity of the workpiece and process-induced forces can also influence size variation. Workpiece Variations in workpiece initial stock conditions are a common source of output variation. Workpieces should be checked for incoming size and hardness. Both parameters cause changes in process forces. Cpk of incoming parts would be desirable. 8