Process Capability Analysis--Normal and General Non-Normal Distribution (Raw Data) - Advanced, Non-Normal Tab

Select the Advanced, non-normal tab of the Process Capability Analysis--Normal and General Non-Normal Distribution (for Raw Data) dialog box to access the options described here.

Summary: Current variable

Click the Summary: Current variable button to create a spreadsheet containing the normal and non-normal process capability indices.

If the standard 3 time Sigma limits are used for the computations, then the capability indices for the Pearson distributions fit as well as the Johnson distributions fit will be reported. The computation of these indices is described in the Overview; Clements (1989) also provides detailed step-by-step instructions of how these indices are computed.

Note: Pearson distributions vs. Johnson distributions

As described in the Overview, the capability indices based on these two methods for fitting non-normal distributions will in most cases yield very similar results.

To reiterate, you should remember that the estimated four moments (for the non-normal distribution fitting) are subject to potentially substantial sampling fluctuations (Hahn and Shapiro, 1967). Therefore, the capability indices reported here should always be interpreted with caution. Note that for distributions with extreme skewness and/or kurtosis, the capability indices based on the Johnson distributions fit will usually be more conservative (smaller) than those based on the Pearson distributions fit; for distributions with an absolute skewness that is greater than 2, only the capability estimates based on the Johnson distributions fit are reported.

All variables

Click the All variables button to create a summary spreadsheet with the non-normal (Johnson-curves) process capability indices for each variable that you have specified in your design (see also Fitting Distributions by Moments).

In the results spreadsheet, the variables will be the cases; the process capability indices along with additional statistics will be the columns. Note that the All variables button is only available if more than one variable was selected for the analysis via the

Process Capability Analysis Setup--Raw Data

Summary histogram

Click the Summary histogram button to produce a standard process analysis histogram.

However, instead of a normal distribution fit, the fitted non-normal distribution curve is shown in the graph. As described in the Overview, the fitted distribution curve is computed from the respective Johnson transformation of a normal variable, and not from the tabulated values of the Pearson distributions. You can change the options and scaling for the histogram via the

Options

This tab contains an option to display the equivalent percentile values for the respective non-normal distribution instead of the Sigma limits for the normal distribution.

For example, instead of indicating the upper process range as 3 times Sigma, the graph may indicate the location on the x-axis of the graph that corresponds to the 99.865 percentile value (which is the equivalent to 3 times Sigma for the normal distribution). Given that the non-normal fit is appropriate, this option allows you to gain a more realistic picture of the process performance relative to the engineering specifications.

Non-normal frequency distribution

Click the Non-normal frequency distribution button to display a spreadsheet with a frequency table for the observed data.

n addition, the expected values for the respective fitted non-normal distribution will also be reported. Note that the Chi-square goodness-of-fit test has 2 degrees of freedom less than the Chi-square test for the normal distribution because both the skewness and kurtosis were estimated from the data.

Q-Q Non-normal distribution

Click the Q-Q Non-normal distribution button to produce a Q-Q plot.

This type of plot is described in the Introductory Overview (see also Hahn and Shapiro, 1967). To produce a Q-Q plot, Statistica will first sort the N observed data points into ascending order, so that:

x₍₁₎ <=.x₍₂₎ <= ... <=.x_(n)

These observed values are plotted against one axis of the graph; on the other axis the plot will show:

F^-1 ((i-r_adj )/(n+n_adj ))

where i is the rank of the respective observation, r_adj and n_adj are adjustment factors (<= 0.5) and F^-1 denotes the inverse of the probability integral for the respective standardized distribution (i.e., non-normal). The resulting plot is a scatterplot of the observed values against the (standardized) expected values, given the respective distribution. If the respective theoretical distribution provides a good fit to the observed data, then all data points in this plot will fall onto a straight line. Note that Statistica also computes the respective percentile values for the theoretical distribution and places them on the scale opposite the standardized theoretical distribution values.

P-P Non-normal distribution

 Click the P-P Non-normal distribution button to produce a P-P plot.

This type of plot is described in the Introductory Overview (see also Hahn and Shapiro, 1967). In the P-P (probability-probability) plot, the observed cumulative distribution function is plotted against the theoretical cumulative distribution function for the same values. If the respective (non-normal) distribution provides a good fit to the observed data, then all data points will fall onto the diagonal in this plot.