Walker Runn Earns an F (Value)
There’s more to SPC than the program you use to calculate it.

Michael J. Cleary, Ph.D.

Paint manufacturer Color in a Can, along with its evasive quality manager, Walker Runn, is about to discover the pot of SPC at the end of the rainbow.

Runn, caught in a significant statistical error, was sidelined, so to speak, and told to take some time to learn a little more about statistical process control. The perfect teacher was none other than Dan Druff, the vice president who uncovered Walker’s incompetence in the first place. Druff himself is a highly skilled statistician (though he’s known to be a little flaky), and Runn has been following him around to learn enough to keep his job.

After several months of this apprenticeship, Druff gives Runn a chance to demonstrate what he’s learned by producing an analysis of variance, which follows:

“No problem,” Walker says, realizing that even without fully understanding the data, he can simply enter them into an easy-to-use software program for the calculations, then slip the results into a PowerPoint presentation he’s preparing for Druff’s review.

Druff carefully goes over the analysis of variance, also reviewing for Runn what he’d learned from his disastrous hypothesis-testing assumptions several weeks earlier. Druff wants to be sure that Runn understands hypothesis testing as well as ANOVA, so he begins to question his understudy about the F value of 4.114 that’s indicated on the ANOVA output.

“F stands for facilities,” Walker asserts confidently, sure that the chart is self-explanatory and bears no deep statistical secrets. “Color in a Can has three facilities, or F’s,” he continues. It’s clear from Duff’s ominous silence that this can’t be the correct response, and Runn decides he’d better walk or run to the nearest employment agency after this debacle.

Can you define F value? Pick the correct response:

a) F value refers to the ratio of variation between samples to the variation within samples.

b) F value refers to the spread in the control limits for all the data.

Answer “a” is, of course, correct.

“Variance within” is essentially equal to the average variance of the three samples (or in this case, the three facilities).

Variance within = (also known as SSW, sum of squares within)

m = number of samples (3, in this case)

Variance between = (also known as SSB, sum of squares between)

Where nj = size of the jth sample (in this case, 6)

Xj = mean of the jth sample (in this case, 12.17, 13.67, 9)

n = total of all samples (18 in this case)

X = the average of all the data (11.61 in this case)

Now we’re ready to test the hypothesis that the three facilities are essentially the same.

Step 1:

H0 = µ1 = µ2 = µ3

All three lines are essentially the same with respect to output; this constitutes the null hypothesis.

H1 = otherwise

One or more of the lines is different from the others (this is the alternative hypothesis).

Step 2:

a = 0.05

Interpretation: One wants a 5 percent probability of rejecting the null if it’s actually true. This is known as a type 1 error.

Step 3:

Calculate the statistical F value by taking a ratio of the variance between to the variance within, divided by the appropriate degrees of freedom.

Step 4:

Compare the calculated F value to the tabular F value in the back of a statistics textbook. The calculated F value (4.114) is greater than the tabular F value (3.68), so the null hypothesis is rejected.

You may have noticed a p value of 0.038 next to the F value. What does this mean? If you know, e-mail me for a chance to win a copy of DOEpack from PQ Systems Inc. (www.pqsystems.com).

About the author

Michael J. Cleary, Ph.D., is president of PQ Systems Inc.