Chi-Square Critical Value Table
Chi-square critical values by degrees of freedom and alpha, for goodness-of-fit, independence and variance tests, with how to get df for each test type.
Chi-Square Critical Value Table (Right and Left Tail)
For a goodness-of-fit, independence, or variance test, the chi-square critical value is the threshold that defines the rejection region for the null hypothesis. At a significance level α=0.05 with 5 degrees of freedom, that threshold is 11.070 (source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7). You compare this number to your test statistic: if the statistic exceeds 11.070, you reject the null. The most common failure mode is using the wrong tail: chi-square is asymmetric, so the upper critical value is the one for hypothesis tests, not the lower value near zero. This table gives the exact numbers for both tails and explains which to use.
Chi-Square Table (Right Tail)
The right-tail chi-square critical value table gives the threshold for hypothesis tests: goodness-of-fit, test of independence, and test of homogeneity. These tests all use the upper tail of the chi-square distribution. The value depends on the degrees of freedom and the chosen alpha.
The table below is generated from the inverse chi-square function CHISQ.INV.RT in Excel and Google Sheets, which takes a right-tail probability and degrees of freedom. For a test at α=0.05 with df=5, CHISQ.INV.RT(0.05, 5) returns 11.070. At α=0.01, the same df gives 15.086. The NIST/SEMATECH e-Handbook confirms these values to three decimal places.
The choice between right-tail and left-tail is not ambiguous: every hypothesis test for goodness-of-fit, independence, or variance (where the alternative is "greater than") uses the right-tail critical value. Only confidence intervals for variance and certain two-tailed variance tests use the left tail.
| Degrees of Freedom (df) | α = 0.10 | α = 0.05 | α = 0.01 |
|---|---|---|---|
| 1 | 2.706 | 3.841 | 6.635 |
| 2 | 4.605 | 5.991 | 9.210 |
| 3 | 6.251 | 7.815 | 11.345 |
| 4 | 7.779 | 9.488 | 13.277 |
| 5 | 9.236 | 11.070 | 15.086 |
| 6 | 10.645 | 12.592 | 16.812 |
| 7 | 12.017 | 14.067 | 18.475 |
| 8 | 13.362 | 15.507 | 20.090 |
| 9 | 14.684 | 16.919 | 21.666 |
| 10 | 15.987 | 18.307 | 23.209 |
| 15 | 22.307 | 24.996 | 30.578 |
| 20 | 28.412 | 31.410 | 37.566 |
| 30 | 40.439 | 43.773 | 50.892 |
Left Tail and Two-Tailed Values (Variance Tests and Intervals)
Confidence Intervals for Variance
Left-tail chi-square critical values are used for confidence intervals on a population variance and for certain two-tailed variance tests. Because the chi-square distribution is asymmetric and bounded at zero, the left-tail critical value is a small number near zero, not the negative of the right-tail value. For a 95% confidence interval on variance with df=5, you need two chi-square critical values: the left-tail at 0.975 (χ²_0.975,5) and the right-tail at 0.025 (χ²_0.025,5). The left-tail value for df=5 and α/2=0.025 is 0.831, while the right-tail value is 14.733 (from CHISQ.INV.RT(0.025,5) = 14.733).
Two-Tailed Hypothesis Tests
For a two-tailed hypothesis test on variance (H₀: σ² = σ²₀ vs H₁: σ² ≠ σ²₀), you split alpha equally: the lower critical value is χ²_{1−α/2, df} and the upper is χ²_{α/2, df}. At α=0.05 and df=5, the lower critical value is χ²_0.975,5 = 0.831 and the upper is χ²_0.025,5 = 14.733. If your test statistic falls below 0.831 or above 14.733, reject the null. This is the same logic as a two-tailed Z or t test, but the asymmetry means the two boundaries are not symmetric around the mean.
Finding Left-Tail Values
The NIST/SEMATECH e-Handbook provides left-tail chi-square critical values in its tables of quantiles. Software functions like CHISQ.INV.RT in Excel give the right-tail directly; for the left-tail, use CHISQ.INV (which takes a left-tail probability) or compute 1−α for the right-tail function.
Degrees of Freedom Chi-Square Test of Independence
The degrees of freedom for a chi-square test of independence follow a formula based on the number of rows and columns in the contingency table. For a test of independence between two categorical variables, df = (number of rows − 1) × (number of columns − 1). A 3×4 table gives df = (3−1)(4−1) = 6. For a goodness-of-fit test with k categories, df = k − 1 − (number of estimated parameters). For a test of homogeneity across multiple groups, the df calculation follows the same rule as independence: (rows−1)×(cols−1), where rows are the groups and columns are the response categories.
OpenIntro Statistics (4th ed.) gives these formulas as standard convention. Miscomputing df is one of the most common errors in chi-square tests, because the df affects the critical value directly: at α=0.05, df=1 gives a critical value of 3.841, but df=5 gives 11.070. Using the wrong df selects the wrong threshold.
Worked Example: Goodness-of-Fit Test
Scenario: You are testing whether a die is fair. You roll it 60 times, counting the number of 1s, 2s, ... , 6s. Expected count under fairness is 10 per face. The test statistic is χ² = Σ (observed − expected)² / expected = 8.4.
Step 1: Determine df. k = 6 categories, no parameters estimated from the data, so df = 6 − 1 = 5.
Step 2: Choose alpha and find the critical value. At α = 0.05 and df = 5, the chi-square critical value from the right-tail table is 11.070.
Step 3: Compare. Test statistic (8.4) < critical value (11.070). Fail to reject the null hypothesis. The die shows no statistically significant evidence of bias at the 0.05 level.
Failure case: If you had used a left-tail critical value (0.831 for df=5, α=0.05), you would compare 8.4 to 0.831, see that 8.4 > 0.831, and incorrectly reject the null. Always use the right-tail critical value for hypothesis tests. The left-tail value is for confidence intervals on variance.
Degrees of Freedom Formula Box
Degrees of freedom for chi-square tests:
- Goodness-of-fit: df = k − 1 − m, where k = number of categories, m = number of parameters estimated from the sample.
- Test of independence: df = (r − 1)(c − 1), where r = number of rows in the contingency table, c = number of columns.
- Test of homogeneity: df = (r − 1)(c − 1), with r = number of groups, c = number of response categories.
- One-sample variance test: df = n − 1, where n = sample size.
These formulas apply to the chi-square critical value for the right-tail (hypothesis test) and left-tail (confidence interval) tables. The df for a chi-square distribution is always a positive integer.
Left Tail Chi-Square: When to Use It
The left-tail chi-square critical value is used only for confidence intervals on a population variance and for two-tailed variance tests where the alternative is "not equal to." It is not used for goodness-of-fit, independence, or homogeneity tests. For a 95% confidence interval on variance, you need the left-tail critical value at α/2 and the right-tail value at α/2. With df=5, the left-tail 0.025 quantile is 0.831 and the right-tail 0.025 quantile is 14.733. The interval's lower bound is (n−1)s²/χ²_{0.025,5} and the upper bound is (n−1)s²/χ²_{0.975,5}.
Software functions: In Excel and Google Sheets, CHISQ.INV(probability, df) returns the left-tail quantile. CHISQ.INV(0.025, 5) returns 0.831. The CHISQ.INV.RT function gives the right-tail quantile. The TI-84 Plus CE does not have a native chi-square inverse function (as of OS 5.6), so you must use the chi-square cdf and solve manually or use a different tool. The NIST/SEMATECH e-Handbook tables for left-tail chi-square critical values are available in section 1.3.6.7.
Common Questions
What is the chi-square critical value for a goodness-of-fit test with 8 categories at α=0.05?
df = 8 − 1 = 7 (assuming no parameters estimated from the sample). The right-tail chi-square critical value at α=0.05, df=7 is 14.067. Compare your test statistic to 14.067; reject the null if the statistic exceeds this threshold.
When do I use the left-tail chi-square critical value instead of the right-tail?
Use the left-tail value for confidence intervals on a population variance and for two-tailed variance tests where the alternative is "not equal to." For hypothesis tests of goodness-of-fit, independence, or homogeneity, always use the right-tail value.
How do I find a chi-square critical value on a TI-84?
The TI-84 Plus CE does not have a built-in chi-square inverse function (as of OS 5.6). Use the chi-square cdf and trial-and-error to find the value, or use Excel/Google Sheets with CHISQ.INV.RT (right-tail) or CHISQ.INV (left-tail).
What is the chi-square critical value for a two-tailed variance test at α=0.10 with df=10?
Split alpha equally: α/2 = 0.05. The right-tail critical value at α=0.05, df=10 is 18.307. The left-tail critical value at α=0.05, df=10 is 3.940. Both boundaries are needed for the two-tailed test; the rejection region is below 3.940 or above 18.307.
Why does the chi-square critical value change with degrees of freedom?
The chi-square distribution's shape depends on df. With low df (e.g., 1 or 2), the distribution is highly skewed right, so the right-tail critical value is relatively small. As df increases, the distribution becomes more symmetric and the critical values for a given alpha increase. For example, at α=0.05, df=1 gives 3.841, df=10 gives 18.307, and df=30 gives 43.773.