Critical Value

Critical values are used in hypothesis testing to delimitate two regions: one in which the analyst rejects the null hypothesis and another in which the hypothesis cannot be rejected. Therefore, critical values are closely linked to the concept of hypothesis testing. For instance, in a two-sided t test to compare two sample means, if the sample sizes are 21 and 21, so that we have 40 df, the critical value for a 0.05 level of significance (i.e., a 95% level of confidence) is 2.01 (this value can be found in tables for the Student's t test). If the computed t statistic is, say, 2.32, we reject the null hypothesis—that the two samples correspond to two populations with the same population mean—at the 0.05 level of significance, because the computed t value exceeds the critical value (and that implies that if the null hypothesis were true, the probability of values like those observed or more extreme is below 0.05). However, we cannot reject the null hypothesis at the 0.01 level of significance because for that level of significance the critical value is 2.42, which is above the computed t = 2.32.

This example shows that critical values are arbitrary. They are just values corresponding to particular probabilities arbitrarily chosen (usually 0.05 or 0.01) in theoretical statistical distributions. The presentation of confidence intervals is usually preferred to the indication of the fact that the null hypothesis has or has not been rejected because the test statistics was above or below the critical value.

Since hypothesis testing is usually ignored in the Bayesian approach to statistics, the concept of critical value is irrelevant in that theoretical framework.