Capability indices for nonnormal data

Capability indices cannot be estimated the same way for normal and nonnormal data because their distributions are different. For example, the shapes of nonnormal distributions are most likely asymmetric, and the distribution coverage of a nonnormal distribution cannot be represented by the number of standard deviations (a parameter unique to the normal distribution). To calculate capability indices for nonnormally distributed data, equivalent methods are necessary that are analogous to the normal case.

By default, Minitab calculates overall capability indices for nonnormal data by using the Z-score method. First, Minitab calculates the proportion of observations that are outside the specification limits based on the nonnormal distribution that you specified for the analysis. Minitab then uses these proportions to determine the corresponding Z values in the standard normal distribution, denoted as Z.USL and Z.LSL. Z.USL is the corresponding Z value for the proportion of measurements not greater than the upper specification limit, and Z.LSL is for the proportion of measurements not greater than the lower specification limit. The difference between Z.USL and Z.LSL represents the tolerance interval formed by the two specification limits in the standard normal scale. The capability indices are then calculated using the tolerance interval in the standard normal scale and the process spread 6 from a standard normal distribution, which captures 99.74 percent of the process measurements. To see how this method is used to calculate each specific index, go to Using the Z-score method to determine overall capability for nonnormal data.

Another recognized method for estimating overall capability for nonnormal data is to use the 0.135 and 99.865 percentiles (which correspond to the 6 standard deviation spread in the normal case), and compare the specification limits with these percentiles. This method (ISO) is also available in MInitab.

To estimate the probabilities or the percentiles properly for overall capability, we need enough data to estimate the distribution function. A few observations within a subgroup would not work because, with two or more nonnormal parameters to estimate (depending on the distribution), too few observations will yield estimates with large error and consequently inaccurate capability indices.

Therefore, if data follow a nonnormal distribution, assume that the entire data set is from one distribution, and estimate the distribution parameters with all the observations. This approach yields overall capability indices, which only measure the actual performance of the products or process.

If you must estimate within-subgroup variation with nonnormal data, you can enter data from a single subgroup into a worksheet and perform nonnormal capability analysis. A large number of observations, say 30 or more, is better. The resulting overall capability indices will represent the within-subgroup variation of the single subgroup that you entered.