Chi-Square (Χ²) Table | Examples & Downloadable Table

The chi-square (Χ2) distribution table is a reference table that lists chi-square critical values. A chi-square critical value is a threshold for statistical significance for certain hypothesis tests and defines confidence intervals for certain parameters.

Chi-square critical values are calculated from chi-square distributions. They’re difficult to calculate by hand, which is why most people use a reference table or statistical software instead.

Download chi-square table (PDF)

When to use a chi-square distribution table

You will need a chi-square critical value if you want to:

Chi-square distribution table (right-tail probabilities)

Use the table below to find the chi-square critical value for your chi-square test or confidence interval or download the chi-square distribution table (PDF).

The table provides the right-tail probabilities. If you need the left-tail probabilities, you’ll need to make a small additional calculation.

Chi-square distribution table

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How to use the table

To find the chi-square critical value for your hypothesis test or confidence interval, follow the three steps below.

Example: A chi-square test case study
Imagine that the security team of a large office building is installing security cameras at the building’s four entrances. To help them decide where to install the cameras, they want to know how often each entrance is used. They randomly sample 500 people inside the building and ask them which entrance they used to enter the building.

The team wants to use a chi-square goodness of fit test to test the null hypothesis (H0) that the four entrances are used equally often by the population.

To know whether to reject their null hypothesis, they need to compare the sample’s Pearson’s chi-square to the appropriate chi-square critical value.

Step 1: Calculate the degrees of freedom

There isn’t just one chi-square distribution—there are many, and their shapes differ depending on a parameter called “degrees of freedom” (also referred to as df or k). Each row of the chi-square distribution table represents a chi-square distribution with a different df.

You need to use the distribution with the correct df for your test or confidence interval. The table below gives equations to calculate df for several common procedures:

Test or procedure Degrees of freedom (df) equation
Test of a single variance

Confidence interval for variance or standard deviation

df = sample size − 1
Chi-square goodness of fit test df = number of groups − 1
Chi-square test of independence df = (number of variable 1 groups − 1) * (number of variable 2 groups − 1)
McNemar’s test df = 1
Example: Calculating the degrees of freedom
The security team categorized people into four groups in their sample—one group for each entrance. The formula for the chi-square goodness of fit test is:

df = number of groups − 1

df = 4 − 1

df = 3

Step 2: Choose a significance level

The columns of the chi-square distribution table indicate the significance level of the critical value. By convention, the significance level (α) is almost always .05, so the column for .05 is highlighted in the table.

In rare situations, you may want to increase α to decrease your Type II error rate or decrease α to decrease your Type I error rate.

To calculate a confidence interval, choose the significance level based on your desired confidence level:

α = 1 − confidence level

The most common confidence level is 95% (.95), which corresponds to α = .05.

Example: Choosing a significance level
The security team follows convention, choosing a significance level of .05.

Step 3: Find the critical value in the table

You now have the two numbers you need to find your critical value in the chi-square distribution table:

  • The degrees of freedom (df) are listed along the left-hand side of the table. Find the table row corresponding to the degrees of freedom you calculated.
  • The significance levels (α) are listed along the top of the table. Find the column corresponding to your chosen significance level.
  • The table cell where the row and column meet is your critical value.
Example: Finding the critical value in the table
Where the row for df = 3 and the column for α = .05 meet, the critical value is 7.815.

The security team can now compare this chi-square critical value to the Pearson’s chi-square they calculated for their sample. If the critical value is larger than the sample’s chi-square, they can reject the null hypothesis.

Chi-square table critical value

Left-tailed and two-tailed probabilities

The table provided here gives the right-tail probabilities. You should use this table for most chi-square tests, including the chi-square goodness of fit test and the chi-square test of independence, and McNemar’s test.

If you want to perform a two-tailed or left-tailed test, you’ll need to make a small additional calculation.

Left-tailed tests

The most common left-tailed test is the test of a single variance when determining whether a population’s variance or standard deviation is less than a certain value.

To find the critical value for a left-tailed probability in the table above, simply use the table column for 1 − α.

Example: Left-tailed test
Imagine that you have a part-time job making cookies at a bakery. The bakery owner tells you that their cookies usually vary in size with a standard deviation of only 0.2 inches in diameter.

You pride yourself on making every cookie the same size, so you decide to randomly sample 25 of your cookies to see if their standard deviation is less than 0.2 inches.

This is a left-tailed test because you want to know if the standard deviation is less than a certain value. You look up the left-tailed probability in the right-tailed table by subtracting one from your significance level: 1 − α = 1 − .05 = 0.95.

The critical value for df = 25 − 1 = 24 and α = .95 is 13.848.

If the chi-square of your sample is greater than this critical value, you can reject the null hypothesis that your cookies have a standard deviation of 0.2 inches in diameter.

Two-tailed tests

The most common left-tailed test is the test of a single variance when determining whether a population’s variance or standard deviation is equal to a certain value.

A two-tailed test has two critical values. To find the critical values, use the table columns for \dfrac{\alpha}{2} and 1-\dfrac{\alpha}{2}.

Example: Two-tailed test
A children’s clothing manufacturer wants to design their baby hats so that they will fit any six-month-old baby within two standard deviations of the mean head diameter.

They find in a medical textbook that the standard deviation of head diameter of six-month-old babies is 1 inch, but they want to confirm this number themselves. They randomly sample 20 six-month-old babies and measure their heads.

This is a two-tailed test because they want to know if the standard deviation is equal to a certain value. They should look up the two critical values in the columns for:

  • \dfrac{\alpha}{2}=\dfrac{.05}{2}=.025 and
  • 1-(\dfrac{\alpha}{2})=1-(\dfrac{.05}{2})=1-.025=.975

The critical value for df = 20 − 1 = 19 and α = .025 is 32.852. The critical value for df = 19 and α = .975 is 8.907.

If the chi-square of their sample is not between these two critical values, the clothing company can reject the null hypothesis that the standard deviation of head diameter is 1 inch.

Practice questions

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Frequently asked questions about chi-square tables

How do I find a chi-square critical value in R?

You can use the qchisq() function to find a chi-square critical value in R.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05:

qchisq(p = .05, df = 22, lower.tail = FALSE)

How do I find a chi-square critical value in Excel?

You can use the CHISQ.INV.RT() function to find a chi-square critical value in Excel.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05, click any blank cell and type:

=CHISQ.INV.RT(0.05,22)

What properties does the chi-square distribution have?

A chi-square distribution is a continuous probability distribution. The shape of a chi-square distribution depends on its degrees of freedom, k. The mean of a chi-square distribution is equal to its degrees of freedom (k) and the variance is 2k. The range is 0 to ∞.

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Shaun Turney

During his MSc and PhD, Shaun learned how to apply scientific and statistical methods to his research in ecology. Now he loves to teach students how to collect and analyze data for their own theses and research projects.