A researcher would like to find out whether a man's nickname affects his cholesterol reading (though it is not clear why she believes it should). She records the cholesterol readings of 23 men nicknamed Sam, 24 men nicknamed Lou and 19 men nicknamed Mac; her data appears in the table to the right. She wants to know whether the differences in the average readings are significant; i.e., whether the average reading of all men nicknamed Sam is different from the average reading of all Lous or whether these averages differ from the average reading of all Macs.
SamLouMac
364260156
245204438
284221272
172285345
198308198
239262137
259196166
188299236
256316168
263216269
329155296
136212236
272201275
245175269
209241142
298233184
342279301
217368262
358413258
412240
382243
593325
261156
280

Using the ANOVA formulas, we have:

• I = 3;
• n1 = 23, n2 = 24, and n3 = 19, and x1,1 = 364, x1,2 = 260, x1,3 = 156, x2,1 = 245, etc.;
• N = 23 + 24 + 19 = 66;
• AV1 = 283.6, AV2 = 253.7, AV3 = 242.5 (at least approximately -- all figures except the degrees of freedom are approximate from here on);
• AV = [23(283.6) + 24(253.7) + 19(242.5)]/66 = 260.9;
• SSE = (364 - 283.6)2 + (260 - 253.7)2 + (156 - 242.5)2 + (245 - 283.6)2 + . . . (a total of 66 terms) = 406,259.7;
• SSG = 23(283.6 - 260.9)2 + 24(253.7 - 260.9)2 + 19(242.5 - 260.9)2 = 19,485.3;
• DFE = 66 - 3 = 63, and so MSE = 6448.6;
• DFG = 3 - 1 = 2, and so MSG = 9742.7; and hence, finally,
• F = 1.51.
And by consulting an F-distribution table with DFG = 2 and DFE = 63, we find that the probability of a F-value of at least is about 23%, not small enough to conclude statistical significance. We do not reject the null hypothesis and conclude that the average cholesterol readings of all Sams, all Lous and all Macs are not different.

Here is a screenshot of an Excel spreadsheet in which these computations are done.