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Chi-Square Table

By Jim Frost 2 Comments

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.

Annotated chi-square distribution with a critical value and probability.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.

Chi-square table of critical values.

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!

Learn more about how chi-square tests work, significance levels, and degrees of freedom in hypothesis tests.

Tables for other statistics include the z-table, t distribution table, and F-table.

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.

Example of finding a right-tail critical value in a chi-square table.

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.

Probability plot of chi-square distribution with right-tailed probability.

Related post: Test Statistics and One- vs. Two-Tailed Hypothesis Tests

Left-Tailed Tests

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.

Example of finding a left-tailed critical value using a chi-square 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.

Probability plot of a chi-square distribution with the left-tail probability.

Two-Tailed Tests

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.

Example of finding two-tailed critical values using a chi-square table.

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.

Probability plot with a chi-square distribution with two critical regions.

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Filed Under: Hypothesis Testing Tagged With: distributions, graphs

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Comments

  1. nabbona vicky says

    April 12, 2022 at 11:50 pm

    Hello can you pliz tell me how to write the hypothesis statistically

    Reply
  2. Bahjat says

    March 25, 2022 at 6:33 am

    Hello. Could you please give practical example on when or why do we need left tailed chi square test? I understand that the critical value in the right tail helps us to reject the null hypothesis by comparing the calculated χ2 value with it so if the χ2 is equal or greater than the critical value we reject the null hypothesis and if χ2 is lower than the critical value we fail to reject it, so, what does the critical value on the left tail indicate?

    Reply

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