Capability for Non-Normal Data HEATING, COOLING & WATER HEATING PRODUCTS Capability for Non-Normal Data

Normality and Non-Normality The calculated capability indices and the corresponding % out-of-spec values are only valid when the individual data points are normally distributed. Special causes tend to distort the Normal curve. Let’s look at some reasons that will explain non-normality.

Consequences of Estimating Capability Using Non-Normal Data Process Capability Consequences of Estimating Capability Using Non-Normal Data Example: Calculating Process Sigma with a Normal curve To determine Process Sigma, find the defect area beyond the specification limits If the data is not Normal, the defect area will be incorrectly estimated This will affect capability estimates as well

Handling Process Capability Situations Where the Underlying Distribution is Known to be Non-Normal Consider the following flatness data. This data is obviously not normal, and would not be expected to be so…. Let’s look at the capability analysis for the raw data.

Use a Normal Probability Plot to Check for Normality Process Capability Use a Normal Probability Plot to Check for Normality Here is a sample Normal probability plot generated in Minitab (n = 25). Graph > Probability Plot If the data are Normal, the points will fall on a “straight” line. “Straight” means within the 95% confidence bands. You can say the data are Normal if approximately 95% of the data points fall within the confidence bands. 95% confidence bands

What Is a Normal Probability Plot? Process Capability What Is a Normal Probability Plot? 20 30 10 70 80 90 50 10% Ten equally spaced percentiles from the Normal distribution Normal Probability Plot Data values are on X-axis Percentiles of the Normal distribution are on the Y-axis (unequal spacing of lines is deliberate) Equally spaced percentiles divide the Normal curve into equal areas The percentiles match the percents on the vertical axis of the Normal probability plot

Conclusions From Two Normal Probability Plots Process Capability Conclusion Not a serious departure from Normality Conclusion There is a serious departure from Normality

Click: Stat Quality Tools Capability Analysis Normal Click the Lower Spec Boundary box OK Note the Cpk and Ppk values (3.10 & 2.72). Because this distribution is not normal, the capability indices are not correct.

Click: Stat Quality Tools Capability Analysis Nonnormal Click the Johnson transformation box Click the Lower Spec Boundary box OK Note the Ppk value (1.60). Cpk’s are not available with this procedure. Note that an equation for the transformed data is provided.

Distribution Identification in Minitab Click: Stat Quality Tools Individual Distribution Identification Click the Use all distributions and transformations box OK Scan the various distributions (there will be 4 screens containing the normality plots of 16 distribution models to find the highest p-value and the normal plot that looks closest to normality

This screen shows that the Johnson Transformation has the distribution with the highest p-value and best normality plot. Look in the session window to find this equation for the transformed data. 2.19576 + 0.824215 * Ln [ ( X + 0.0000224433 ) / ( 0.0105373 - X ) ] Note: Ln = Natural Log

You can then use the equation to compute a transformed value for each data point. Here is a histogram of the transformed data. If you want to do the capability analysis on the transformed data, you need to use the equation to transform the boundary value and spec limit as well. Let’s run the capability analysis on the transformed data.

Capability Analysis of the transformed data: Click: Stat Quality Tools Capability Analysis Normal Click the Lower Spec Boundary box Add the calculated values for the boundary and USL OK Note the Ppk value (1.60) is the same. Cpk’s are available with this procedure, although the procedure is much more difficult to handle.

Here is a control chart of the transformed data: If you need just a control chart for non-normal data but don’t need any capability analysis information, use the Box-Cox transformation procedure.

Box-Cox Transformation Click: Stat Control charts Box-Cox transformation Set subgroup size Options Select a column to store the transformed data OK in each box Disregard the resulting plot Run a control chart on the new data Notice that although the numerical limits are different then the previous charrt, the pattern is the same as the control chart generated from the Johnson Transformation.

Additional Information on Transformations