Page 182 - Statistics II for Dummies

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Part III: Analyzing Variance with ANOVA
You can find MST by taking SST and dividing by k – 1 (where k is the
number of treatments).
✓ MSE is the mean sums of squares for error, which measures the mean
within-treatment variability. The within-treatment variability is the
amount of variability that you see within each sample itself, due to
chance and/or other factors not included in the model.
You can find MSE by taking SSE and dividing by n – k (where n is the
total sample size and k is the number of treatments). The values of
k – 1 and n – k are called the degrees of freedom (or df) for SST and SSE,
respectively.

Minitab calculates and posts the degrees of freedom for SST, SSE, MST, and
MSE in the ANOVA table in columns two and four, respectively.

From the ANOVA table for the seed-spitting data in Figure 9-4, you can see
that column two has the heading DF, which stands for degrees of freedom.
You can find the degrees of freedom for SST in the Factor row (row two); this
value is equal to k – 1 = 4 – 1 = 3. The degrees of freedom for SSE is found to
be n – k = 20 – 4 = 16. (Remember, you have four age groups and five children
in each group for a total of n = 20 data values.) The degrees of freedom for
SSTO is n – 1 = 20 – 1 = 19 (found in the Total row under DF). You can verify
that the degrees of freedom for SSTO = degrees of freedom for SST + degrees
of freedom for SSE.

The values of MST and MSE are shown in column four of Figure 9-4, with the
heading MS. You can see the MST in the Factor row, which is 29.92. This
value was calculated by taking SST = 89.75 and dividing it by degrees of free-
dom, 3. You can see MSE in the Error row, equal to 3.55. MSE is found by
taking SSE = 56.80 and dividing it by its degrees of freedom, 16.

By finding the mean sums of squares, you’ve completed step two of the
F-test, but don’t stop here! You need to continue to the next section in order
to complete the process.

Figuring the F-statistic

The test statistic for the test of the equality of the k population means is
. The result of this formula is called the F-statistic. The F-statistic
has an F-distribution, which is equivalent to the square of a t-test (when the
numerator degrees of freedom is 1; see more on this interesting connection
between the t- and F-distributions in Chapter 12). All F-distributions start at

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