This chi-square table provides the critical values for chi-square (χ2) hypothesis tests. The column and row intersections are the right-tail critical values for a given probability and degrees of freedom.
Typically, use your significance level to choose the column. Rows contain the degrees of freedom for your chi-square test.
Learn how to use this chi-square table using the information, examples, and illustrations below the table.
Related post: What are Critical Values?
How to Use the Chi-Square Table
Use the chi-square table to find the critical value for your test. The column and row intersections are the right-tail critical values for a given significance level and degrees of freedom.
In the chi-square table, its components represent the following:
- Column headings indicate the probability of χ2 ≥ the critical value.
- Row headings define the degrees of freedom for your chi-square test.
- Cells within the table represent the critical chi-square value for a right-tailed test.
In chi-square tests, the degrees of freedom equal: (Number of Columns – 1) * (Number of Rows – 1)
While the table is set up for right-tailed tests, simple procedures allow you to use it for other types of χ2 tests.
In the sections below, learn how to find the critical values for right-tailed, left-tailed, and two-tailed χ2 tests. For the following examples, I include sampling distribution plots that correspond to the values in the example. They help link the plain-looking chi-square tables to something more intuitive!
Chi-square Table: Right-Tailed Tests
The most straightforward problem is finding the right-tail critical value because the chi-square table displays that without further calculations. Right-tailed chi-squared tests are the most common type.
Suppose you use a significance level of 0.05, and your chi-square test has 5 degrees of freedom. The truncated chi-square table below shows the critical χ2 value.
The table indicates that the critical value is 11.070. If the χ2 test statistic is greater than or equal to 11.070, our results are statistically significant. The probability distribution plot below displays this graphically.
Other problems might ask you to use the chi-square table to find left-tail critical values. To find probabilities for areas to the left of a critical value, take your significance level and subtract it from 1. Use that column in the chi-square table to find the left-tail critical value. For example, if your significance level is 0.05, then use 1 – 0.05 = 0.95.
Again, suppose that we use α = 0.05 and have 5 df. However, this time we are performing a left-tailed test. Consequently, we’ll use the 1 – 0.05 = 0.95 column in the table.
The chi-square table displays the critical χ2 value of 1.145. This result tells us that 95% of the values are to the right of this value. Because the total area under the distribution equals 1, there must be 5% to the left of it. Hence, 1.145 is the critical value for our left-tailed test. If our χ2 test statistic is less than or equal to 1.145, the results are statistically significant.
The graph below displays this result visually.
For a two-tailed chi-square test, divide your significance level in half because the test splits the area between the upper and lower tails.
Because chi-square distributions are greater than zero and not symmetrical, its critical values won’t be ± the same value as they are for z and t distributions. Instead, they are two different positive numbers. Consequently, we need to find the right-tail and left-tail critical values separately using the two procedures above.
For this example, we’re performing a two-tailed χ2 test with α = 0.05 and 5 degrees of freedom.
For a significance level of 0.05, we need to find the critical values for the following probabilities:
- Right-tail critical value: 0.05 / 2 = 0.025.
- Left-tail critical value: 1 – (0.05 / 2) = 0.975
In the chi-square table below, I highlight these two results.
The chi-square table shows that our lower critical value is 0.831 and the upper critical value is 12.833. Consequently, our results are statistically significant if the χ2 test statistic when ≤ 0.831 or ≥ 12.833.
The graph displays these results visually.