I would like to find out whether a Man's Name affects his credit score. I recorded the Credit scores readings of 25 men Named Joe, 22 men named Jon and 19 men named Mark; My data appears in the table to the right. I want to know whether the differences in the average readings are significant. Basically whether the average reading of all men named Joes is different from the average reading of all Jon' or whether these averages differ from the average reading of all Marks.
Joe Jon Mark
650 680 650
630 650 570
630 660 580
650 690 520
670 550 420
440 480 650
550 440 660
549 350 670
490 650 660
465 670 650
440 690 570
470 680 550
490 650 680
650 660 690
660 650 700
665 690 650
670 720 660
665 550 670
545 650 660
550 660
590 670
655 650
660
670
680

Using the ANOVA formulas, we have:
• I = 3;
• n1 = 26, n2 = 23, and n3 = 20, and x1,1 = 650, x1,2 = 680, x1,3 = 650, x2,1 = 630, etc.;
• N = 25 + 22 + 19 = 69;
• AV1 = 594, AV2 = 253.7, AV3 = 242.5 (at least approximately -- all figures except the degrees of freedom are approximate from here on);
• AV = [26(594) + 23(626.52) + 20(626)]/66 = 642.03
• SSE = (650 - 594)2 + (680 – 626.52)2 + (650 - 626)2 + (630-594)2 + . . . (a total of 66 terms) =448591.7
• SSG = 26(594 – 642.03)2 + 23(626.52 – 642.03)2 + 20(626 – 642.03)2 = 2804.35
• DFE = 69 - 3 = 66, and so MSE = 7126796.84
• DFG = 3 - 1 = 2, and so MSG =1402.18 ; and hence, finally,
• F = 1.24.
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 30%, not small enough to conclude statistical significance. We do not reject the null hypothesis and conclude that the average credit ratings of Joe, Jon, and Mark are not different.
Here is a screenshot of an Excel spreadsheet in which these computations are done.

Anova: Single Factor SUMMARY
Groups Count Sum Average Variance
Joe 26 15444 594 6919.6
Jon 23 14410 626.5217 8332.806
Mark 20 12520 626 4856.842

ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 16885.33 2 8442.667 1.242145 0.295426 3.135918
Within Groups 448591.7 66 6796.845 Total 465477.1 68