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# ANOVA

"Analysis of Variance." A statistical test for heterogeneity of means by analysis of group variances. ANOVA is implemented as ANOVA[data] in the Wolfram Language package ANOVA` .

To apply the test, assume random sampling of a variate with equal variances, independent errors, and a normal distribution. Let be the number of replicates (sets of identical observations) within each of factor levels (treatment groups), and be the th observation within factor level . Also assume that the ANOVA is "balanced" by restricting to be the same for each factor level.

Now define the sum of square terms

 (1) (2) (3) (4) (5)

which are the total, treatment, and error sums of squares. Here, is the mean of observations within factor level , and is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtaining the P-value corresponding to the calculated F-ratio of the mean squared values

 (6)
 category freedom SS mean squared F-ratio model SSA error SSE total SST

If the P-value is small, reject the null hypothesis that all means are the same for the different groups.

Factor Level, Least Squares Fitting, MANOVA, Replicate, Variance

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## References

Miller, R. G. Beyond ANOVA: Basics of Applied Statistics. Boca Raton, FL: Chapman & Hall, 1997.

ANOVA

## Cite this as:

Weisstein, Eric W. "ANOVA." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ANOVA.html