I'm very new to this area and am having difficulty understanding the concept of rejecting the null hypothesis based upon results from the ANOVA table.

  • How do the calculated F and the critical value relate to the p-value?

  • And if the calculated F is greater than 1, does that always indicate that the null hypothesis should be rejected, even if the p-value is less than the alpha?

Sorry if these questions are signs of my ignorance, but I'm 57 and returning to school after a 35 year absence! Thanks for any help.


3 Answers 3


Think about if you have 2 friends who are both arguing over which one lives farther from work/school. You offer to settle the debate and ask them to measure how far they have to travel between home and work. They both report back to you, but one reports in miles and the other reports in kilometers, so you cannot compare the 2 numbers directly. You can convert the miles to kilometers or the kilometers to miles and make the comparison, which conversion you make does not matter, you will come to the same decision either way.

It is similar with test statistics, you cannot compare your alpha value to the F-statistic you need to either convert alpha to a critical value and compare the F-statistic to the critical value or you need to convert your F-statistic to a p-value and compare the p-value to alpha.

Alpha is chosen ahead of time (computers often default to 0.05 if you don't set it otherwise) and represents your willingness to falsely reject the null hypothesis if it is true (type I error). The F-statistic is computed from the data and represents how much the variability among the means exceeds that expected due to chance. An F-statistic greater than the critical value is equivalent to a p-value less than alpha and both mean that you reject the null hypothesis.

We don't compare the F-statistic to 1 because it can be greater than 1 due only to chance, it is only when it is greater than the critical value that we say it is unlikely to be due to chance and would rather reject the null hypothesis.

In the classes that I teach I have found that the students who are not quite as young as the others and are returning to school after working for a while often ask the best questions and are more interested in what they can actually do with the answers (rather than just worrying if it is on the test), so don't be afraid to ask.

  • 1
    $\begingroup$ This answer by @GregSnow is very good. I just thought I'd point to the wikipedia page explaining the p-value - the first couple of paragraphs in particular - since understanding that seems to be a particular bugbear. (I'd alo echo his comments regarding older students.) $\endgroup$
    – Glen_b
    Feb 24, 2013 at 9:33
  • 1
    $\begingroup$ Also see statdistributions.com/f . Across many examples, when the 2 variances used to compute F are divided to obtain a ratio, one gets the sort of distribution shown--IF nothing but chance is operating. The question is, how unlikely would a given F be under such an assumption? $\endgroup$
    – rolando2
    Feb 24, 2013 at 12:57

So in short, Reject the null when your p value is smaller than your alpha level. You should also reject the null if your critical f value is smaller than your F Value, you should also reject the null hypothesis.The F value should always be used along with the p value in deciding whether your results are significant enough to reject the null hypothesis. If you get a large f value, it means something is significant, while a small p value means all your results are significant. The F statistic just compares the joint effect of all the variables together. To put it simply, reject the null hypothesis only if your alpha level is larger than your p value.

Source: http://www.statisticshowto.com/f-value-one-way-anova-reject-null-hypotheses/


I had read the post you recommended, however I felt that it had got a problem and I still don't understand. I captured its content and attached as an image bellow. Could you help to explain it clearly? The conflicting explanation

  • $\begingroup$ F critical value is NOT any statistic. Try to find other books to read. $\endgroup$
    – user158565
    Jun 28, 2019 at 4:28

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