An ANOVA example

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.

Sam	Lou	Mac
364	260	156
245	204	438
284	221	272
172	285	345
198	308	198
239	262	137
259	196	166
188	299	236
256	316	168
263	216	269
329	155	296
136	212	236
272	201	275
245	175	269
209	241	142
298	233	184
342	279	301
217	368	262
358	413	258
412	240
382	243
593	325
261	156
	280

Using the ANOVA formulas, we have:

I = 3;
n₁ = 23, n₂ = 24, and n₃ = 19, and x_1,1 = 364, x_1,2 = 260, x_1,3 = 156, x_2,1 = 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.