One-Way Anova is a parametric test used to compare the mean of two or more population.

For run this test it’s important that:

  • The data from the K population are continuous;
  • The data from the K population are normally distributed;
  • The variance of the k population are the same.

The hypothesis of this test is like this:

  • H0: m1 = m2 =m3 = .. = mi that means that all the sample (y1,..,yi) mean and the target mean are equal;
  • H1: at least one of the means are different.

The way to conduct the hypothesis test is like the one-sample t-tests, but we use another statistics, the F-Statistics.

You need to calculate all the value in the eight formula of image1:

 image1  - ANOVA eight formula
image1 – ANOVA eight formula


  • (1) SSF is the sum of the square of the difference between the mean of each sample (factor) (4) and the mean of all the mean (5)
  • (2) SSE it’s basically the sum of all the error square for each sample because for each sample you get the difference between each element of the sample and its mean (4).
  • (6) MSF and (7) MSE is the mean of (1) SSF and (2) SSE
  • (1) F is the F-Statistics that we use for the One Way Anova. Finally you can calculate it as MSF/MSE.
  • Remember that you have i sample and j element in each sample.

To calculate the critical value of the F-statistics, you have two degree of freedom:

  • DF1 = i-1 that came from the number of elementof the SSF
  • DF2 = i*(j-1) that come from the number of element of the SSE

At this point, you have the F-Statistics, the critical value, you have all you need for your hypothesis test.

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