Process Capability 1

Process Capability

Capability Analysis is the ability of a process to meet specifications (customer requirements). For processes with outputs measured on continuous scales, process capability can be measured not only by the Sigma Level, but can also be measured more directly by indices that compare the spread (variability) and centering of the process to the upper and lower specifications.

Capability can be represented by two pairs of indices: (1) Cp and Cpk which use estimated sigma, and (2) Pp and Ppk which use calculated sigma. Both pairs of indices compare the process spread and centering to the process tolerance (the difference between the upper and lower specification).

When is it used?

• Calculated when it is necessary to understand the ability of a process to meet requirements. • Frequently used in the Measure phase of the DMAIC cycle. • Also performed in the Improve phase to verify the effectiveness of actions that have been taken. • Cpk and Ppk are not appropriate capability indices for attribute or discrete data. If you have a direct measure of defect frequency, such as p-bar, c-bar, or X-bar of a count, then that direct measure is the best indication of capability.

Preliminary Steps to Calculating Process Capability

1. The process must be stable, or in control. 2. The shape of the process distribution must be understood. The assumption of normality must be verified through testing. 2 (a) Create a histogram to visually review the data. The CONDITIONS sample data should be randomly selected and large enough to provide a valid representation of the distribution. A sample size of at least 50 is recommended. (b) Use a Chi-square test of “Goodness of Fit” to quantify how close to normal the distribution really is.

The Chi-square Goodness-of-Fit Test for the normal distribution is one of the most straightforward checks of normality. The null and alternative hypotheses for this test are respectively:

H0: The data follow a Normal distribution

H1: The data do not follow a Normal distribution

The test compares the frequencies in the intervals of a histogram to the expected frequencies if the distribution were normal.

In essence, this test involves overlaying the sample data with a normal curve and measuring how well the data fits the curve, as illustrated.

• The difference between the observed and expected (under the normal distribution) frequency are calculated for each interval.

• These differences are then squared and divided by the expected frequency for that interval.

• By summing the resulting numbers, a Chi-square test statistic value is obtained.

The equation for the test statistic is:

(O − E)2 x2 = ∑ E

The test statistic value is then compared to a critical (cut-off) value from the Chi- square distribution at the target significance level and appropriate degrees of freedom. The degrees of freedom are equal to the total number of histogram intervals minus 3 (one Degree of Freedom each used to calculate the mean and standard deviation, plus one). Thus, a downside of this test is that the degrees of freedom depend on the number of histogram intervals. However, for larger samples (50 or more) the test gives reasonably good results.

Methodology The null hypothesis states that the data fit the normal curve (the alternative being that the data deviate from the normal curve to an unacceptable degree). Hence, a large p-value leading to non-rejection of the null hypothesis is desirable. In other words, if the test statistic value is smaller than the critical value at the desired significance level, we fail to reject the null hypothesis and can conclude that for all practical purposes the sample originated from a normally distributed population.

How to Calculate Cp and Cpk

Here is the information you will need to calculate the Cp and Cpk:

X Process average if you are using an I-MR chart, or X Average of averages if you are using an X-bar R chart USL Upper Specification Limit and LSL Lower Specification Limit

σ est Process Standard Deviation

R *The Process Standard Deviation can be calculated directly σ = est from the individual data, or can be estimated. d2

1. Calculate the Cp index to compare the natural six-sigma spread of the process to the tolerance. Cp is often referred to as "Process Potential" because it describes how capable the process could be if it were centered precisely between the specifications. The calculation does not consider where the process is actually centered.

USL − LSL specification Cp = = 6σ est process

2. Calculate the Cpk index to assess process centering in addition to spread. Think of Cpk as a Cp calculation that is handicapped by considering only the half of the distribution that is closest to the specification.

Process capability increases as the process becomes narrower relative to the tolerance, and as the process is centered closer to the target (nominal) value.

If the process is wider than the tolerance, the index is less than 1.0. If the process just fits within the tolerance, the index is 1.0. If the process is narrow enough that it will fit within the tolerance with room to spare, the index is greater than one.

⎪⎧ X − LSL USL − X ⎪⎫ Cpk = min ⎨ , ⎬ 3σ 3σ ⎩⎪ est est ⎭⎪

Exercises:

1. Identify which figure has: ____ Cp > 1; ____ Cp < 1; ____ Cp = 1

2. Which of the following conditions cannot exist?

a. Cp = 1.78, Cpk = -1.78 b. Cp = 2.00, Cpk = 0.00 c. Cp = 1.33, Cpk = 1.50 d. Cp = 1.33, Cpk = 1.33

3. A machining process has a mean of 30.05 in. and a standard deviation of 0.01 in. The client’s specification is 30.00 in + 0.03. What is the Cp and Cpk of this process?

REMEMBER: Cpk and Ppk are alternate ways of expressing the Capability Level - they both reflect the probability of producing a defect.

• Cpk uses the range value from a control chart to estimate the standard deviation, i.e. short term variation or within subgroup. • Ppk calculates the standard deviation directly from the data, i.e. long term variation or within subgroup and between subgroups.