**What is the P-value for in ANOVA?**

In the analysis of variance (ANOVA), we study the variations of between- and within-groups in terms of their respective mean squares (MS) which are calculated by dividing each sum of squares by its associated degrees of freedom. The result, although termed a mean square, is actually a measure of variance, which is the squared standard deviation.

The *F*-ratio is
then obtained as the result of dividing *MS*(between) and *MS*(within). Even if the population means are all equal to
one another, you may get an *F*-ratio
which is substantially larger than 1.0, simply because of sampling error to
cause a large variation between the samples (group). Such *F*-value may get even larger than the *F*-critical value from the *F*-probability
distribution at given degrees of freedom associated with the two *MS* at a
set significant Type I (alpha-) level of error.

Indeed, by referring to the distribution of *F*-ratios with different degrees of
freedom, you can determine the probability of observing an *F*-ratio as large as the one you calculate even if the populations
have the same mean values.

So, the *P*-value is
the probability of obtaining an *F*-ratio
as large or larger than the one observed, assuming that the null hypothesis of
no difference amongst group means is true.

However, under the ground rules that have been followed for many years by inferential statistics, this probability must be equal to, or smaller than, the significant alpha- (type I) error level that we have established at the start of the experiment, and such alpha-level is normally set at 0.05 (or 5%) for test laboratories. Using this level of significance, there is, on average, a 1 in 20 chance that we shall reject the null hypothesis in our decision when it is in fact true.

Hence, if we were to analyze a set of data by ANOVA and our *P*-value calculated was 0.008, which is much smaller than alpha-value of 0.05, we can confidently say that we would be committing just an error or risk of 0.8% to reject the null hypothesis which is true. In other words, we are 99.2% confident not to reject the hypothesis which states no difference among the group means.

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