Chi-Square Distribution Table
Use this chi-square table to find critical values for a chosen number of degrees of freedom and significance level. Select an upper or lower tail, read the table by row and column, and see worked examples for chi-square tests.
df = 5; upper-tail α = 0.05
Critical values in the reference grid are rounded to four decimal places. The selected tail convention determines which cutoff is appropriate.
Chi-Square Critical Values Table
A chi-square critical value is a cutoff determined by degrees of freedom and a chosen tail probability. In the common upper-tail chi-square table, the entry c satisfies P(Χ²_df ≥ c) = α. Larger observed chi-square statistics fall farther into the upper tail for a fixed df.
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Approximate P-Value Bracket from the Table
Compare the observed statistic with the critical values in the selected df row. A coarse table bounds the tail probability; it does not supply an exact p-value.
How to Read a Chi-Square Table
Find the degrees of freedom in the first column, then move to the significance-level column. For df = 5 and upper-tail α = 0.05, the critical value is approximately 11.0705. An observed statistic above this cutoff is in the upper 5% tail of the reference distribution.
Chi-Square Degrees of Freedom
Degrees of freedom depend on the statistical procedure. For a chi-square goodness-of-fit test with k categories and no estimated parameters, df is typically k − 1. For an r-by-c independence table with standard assumptions, df = (r − 1)(c − 1). Other model constraints can change these counts.
Chi-Square Table and P-Values
A printed critical-value table provides thresholds at selected α levels. Comparing a statistic with those thresholds can bracket a p-value, but it usually cannot give its exact value. For an exact upper-tail probability, use a validated chi-square distribution calculator with the correct degrees of freedom.
Upper-Tail vs. Lower-Tail Chi-Square Values
An upper-tail critical value c has probability α above it, while a lower-tail critical value has probability α below it. For df = 5 and α = 0.05, the upper-tail cutoff is about 11.0705 and the lower-tail cutoff is about 1.1455. These are not interchangeable.
Chi-Square Tables in AP Biology and Independence Tests
In classroom goodness-of-fit problems, a chi-square table can help compare an observed test statistic with a critical value at a stated α and df. A contingency table of observed category counts is a separate input used to compute the statistic; it is not the critical-value reference table shown here.
Worked Example: df = 5 at α = 0.05
Choose the upper-tail table, locate row df = 5, and read column α = 0.05. The critical value is χ² ≈ 11.0705. The cutoff is a property of the χ² distribution; whether it applies to a specific hypothesis test depends on that test's assumptions.
Frequently Asked Questions
What is a chi-square table?
A chi-square table lists critical values of chi-square distributions at chosen degrees of freedom and tail probabilities. It is used as a reference for tests whose statistic follows an appropriate chi-square model.
How do I read a chi-square critical value table?
Select the df row and the requested upper- or lower-tail α column. The intersecting entry is the critical χ² value under that convention.
What does df mean in a chi-square table?
df means degrees of freedom. It identifies which chi-square distribution to use and depends on the statistical procedure.
How do I find a p-value from a chi-square table?
Compare your statistic with neighboring critical values at the appropriate df. This gives an interval for the tail probability; an exact p-value requires evaluating the distribution rather than reading only a coarse table.
Is a chi-square critical value table the same as a chi-square test table?
Not necessarily. A chi-square critical table lists reference cutoffs; a contingency table lists observed counts used to calculate a test statistic. They serve different purposes.
Why are lower- and upper-tail chi-square tables different?
The chi-square distribution is asymmetric, so its lower and upper critical values are different and must match the specified tail probability.
How do I use a chi-square table for AP Biology?
First calculate the test statistic and correct degrees of freedom for the assigned test. Then look up the reference critical value for the chosen α. Interpret results in light of the test assumptions.
What is the chi-square critical value for df = 5 at 0.05?
The upper-tail chi-square critical value is approximately 11.0705. The lower-tail value at the same α is about 1.1455.
Sources
- NIST: Critical Values of the Chi-Square Distribution
- OpenStax: Facts About the Chi-Square Distribution
Reviewed: October 2026. Reference grids and lookups are calculated locally; no observations are sent to a server.