The significance level determines how much probability is assigned to results considered sufficiently unusual under the null hypothesis. A one-tailed test places the relevant cutoff in one distribution tail, whereas a two-tailed test uses cutoffs in both tails. Therefore, selecting the correct alpha level and test direction is essential before comparing a calculated statistic with the table.
The table must match the probability distribution associated with the test statistic. The z, t, chi-square, and F distributions provide different cutoff values because they represent different statistical testing contexts. Using the wrong distribution can produce an inappropriate comparison and lead to an incorrect decision about the evidence against the null hypothesis.
Degrees of freedom can determine which cutoff applies, particularly for the t, chi-square, and F distributions. As the relevant degrees of freedom change, the table may provide a different critical value even when the significance level and tail structure remain unchanged. Researchers must therefore identify the correct degrees of freedom before reading the table.
First identify the test statistic's probability distribution, then select the significance level and determine whether the test is one-tailed or two-tailed. If required, locate the appropriate degrees of freedom. Finally, compare the calculated statistic with the selected cutoff and use that comparison to decide whether to reject or retain the null hypothesis.
After calculating a test statistic, researchers use the relevant cutoff to evaluate whether the observed result provides sufficient evidence against the null hypothesis. This approach supports consistent interpretation across experimental and observational studies. Reporting the distribution, significance level, tail structure, and relevant degrees of freedom also makes the decision process more transparent.
Critical values are used not only for hypothesis-testing decisions but also for constructing confidence intervals. The selected cutoff reflects the relevant probability distribution and significance setting, while the resulting interval communicates a range associated with the analysis. This connects the decision about the null hypothesis with an estimate presented in interval form.