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F-Ratio Formula when the groups are the same size

MSbetween and MSwithin estimate the same value (following the belief that H0 is true), then the F-ratio should be approximately equal to one. Mostly, just sampling errors would contribute to variations away from one. As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples. MSwithin is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin.Then the F-ratio will be larger than one. However, if the population effect is small, it is not unlikely that MSwithin will be larger in a given sample.

The foregoing calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the
F-ratio can be written as:

F-Ratio Formula when the groups are the same size


where …

  • n = the sample size
  • dfnumerator = k – 1
  • dfdenominator = n – k
  • s2pooled = the mean of the sample variances (pooled variance)
  • s2¯¯¯xsx¯2 = the variance of the sample means