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25

Components of Variance

KEY WORDS analysis of variance, ANOVA, components of variance, composite sample, foundry
waste, nested design, sampling cost, sample size, system sand.

A common problem arises when extreme variation is noted in routine measurements of a material. What
are the important sources of variability? How much of the observed variability is caused by real
differences in the material, how much is related to sampling, and how much is physical or chemical
measurement error? A well-planned experiment and statistical analysis can answer these questions by
breaking down the total observed variance into components that are attributed to specific sources.

Multiple Sources of Variation
When we wish to compare things it is important to collect the right data and use the appropriate estimate
of error. An estimate of the appropriate error standard deviation can be obtained only from a design in
which the operation we wish to test is repeated.
The kind of mistake that is easily made is illustrated by the following example. Three comparisons
we might wish to make are (1) two testing methods, (2) two sampling methods, or (3) two processing
methods. Table 25.1 lists some data that might have been collected to make these comparisons.
Two Testing Methods. Suppose we wish to compare two testing methods, A and B. Then we must
compare, for example, ten determinations using test method A with ten determinations using test method
B with all tests made on the same sample specimen.
Two Sampling Methods. To compare two different sampling methods, we might similarly compare
determinations made on ten different specimens using sampling method A with those made on ten
specimens using method B with all samples coming from the same batch.
Two Processing Methods. By the same principle, to compare standard and modified methods of
processing, we might compare determinations from ten batches from the standard process with deter-
minations from ten batches made by the modified process.
The total deviation y − η of an observation from the process mean is due, in part, to three sources:
(1) variation resulting only from the testing method, which has error e t and standard deviation σ t ; (2)
variation due only to sampling, which has error e s and standard deviation σ s ; and (3) variation due only
to the process, which has error e p and standard deviation σ p .
Figure 25.1 shows the dot plots of the data from the three testing schemes along with the sample
statistics from Table 25.1. The estimated variance of 0.50 obtained from making ten tests on one specimen
2
provides an estimate of σ t , the testing variance. The variance V s = 5.21 obtained from ten specimens
2
2 alone, but rather σ s + 2
each tested once does not estimate σ s σ t . This is because each specimen value
includes not only the deviation e s due to the specimen but also the deviation e t due to the test. The
variance of e s + e t is σ s + σ t 2 because the variance of a sum is the sum of the variances for independent
2
sources of variation. In a similar way, the variance V p = 35.79 obtained from ten batches each sampled
2
and tested once is an estimate of σ p + σ s + σ t .
2
2
2
Using a “hat” notation (σ ˆ ) to indicate an estimate:
σ ˆ t = 0.50
2
© 2002 By CRC Press LLC