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

2. 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.

7. 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

8. 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.

9. 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

10. 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

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

12. 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.

13. 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.

14. 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

15. 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.
* Ln [ ( X ) / ( X ) ]
Note: Ln = Natural Log

16. 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.

17. 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.

18. 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.

19. 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.

20. Additional Information on Transformations