Walker Runn Earns an F (Value)
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 easytouse 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
hypothesistesting 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 selfexplanatory 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.
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).
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.
Calculate the statistical F value by taking a ratio of
the variance between to the variance within, divided by
the appropriate degrees of freedom.
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, email me for
a chance to win a copy of DOEpack from PQ Systems Inc.
(www.pqsystems.com).
Michael J. Cleary, Ph.D., is president of PQ Systems
Inc.
