A chi-square value is significant when it is large enough that the difference you observed in your data is unlikely to have occurred by random chance alone. There is no single number that qualifies as “significant” across all situations. Instead, significance depends on two things: your degrees of freedom (which reflect the size of your table or number of categories) and the significance level you choose, almost always set at 0.05. Your calculated chi-square value is compared against a critical value from a chi-square distribution table, and if your value exceeds that threshold, the result is statistically significant.
How Chi-Square Significance Works
The chi-square test measures how much your observed data differs from what you would expect if there were no real relationship or difference. The test produces a single number: the chi-square statistic. A small chi-square value means your data closely matches what you’d expect by chance. A large value means something in your data deviates from the expected pattern.
To determine whether that deviation is meaningful, you compare your chi-square statistic against a critical value. If your statistic is greater than the critical value, you reject the assumption that nothing interesting is happening (the null hypothesis) and call the result statistically significant. If it falls below, the observed difference could plausibly be due to chance.
The Role of P-Values and the 0.05 Threshold
Every chi-square statistic has a corresponding p-value, which tells you the probability of seeing a result this extreme if there were truly no effect. A p-value of 0.04, for example, means that if there were genuinely no difference between your groups, you’d see a result this large or larger only 4% of the time across many repeated experiments.
The standard cutoff is p < 0.05. This 5% threshold has a basis in the properties of the normal distribution: roughly 5% (4.5%, to be precise) of values in a normal distribution fall more than two standard deviations from the mean, making them statistical outliers. When your chi-square test produces a p-value below 0.05, you conclude the result is statistically significant. When it’s above 0.05, you don’t have enough evidence to rule out chance.
To illustrate: in a clinical trial comparing a drug to a placebo, if 60% of patients responded to the drug versus 40% to placebo, a chi-square test might produce a p-value of 0.04. Because that falls below 0.05, you’d conclude the drug performed significantly better. If the p-value were 0.08, the same difference in percentages would not reach significance.
Why Degrees of Freedom Change the Threshold
The critical value your chi-square statistic must beat depends on your degrees of freedom. For a simple two-category comparison (one degree of freedom), the critical value at the 0.05 level is 3.841. That means any chi-square statistic above 3.841 is significant. But for a larger analysis with 5 degrees of freedom, the critical value rises to 11.07. With 10 degrees of freedom, it’s 18.31.
Degrees of freedom reflect the complexity of your data. In a contingency table (the kind you use to compare groups across categories), degrees of freedom equal the number of rows minus one, multiplied by the number of columns minus one. A 2×2 table has 1 degree of freedom. A 3×4 table has 6. In a goodness-of-fit test, where you’re checking whether data fits an expected distribution, degrees of freedom equal the number of categories minus one, minus any additional parameters you estimated from the data.
The key takeaway: you cannot look at a chi-square value in isolation and call it significant. A chi-square of 5.0 is significant with 1 degree of freedom but not with 5.
Common Critical Values at the 0.05 Level
These are the chi-square values your result must exceed to be significant at p < 0.05, for the most commonly encountered degrees of freedom:
- 1 degree of freedom: 3.841
- 2 degrees of freedom: 5.991
- 3 degrees of freedom: 7.815
- 5 degrees of freedom: 11.07
- 10 degrees of freedom: 18.31
If you’re using a stricter threshold of p < 0.01, these values increase. For 1 degree of freedom, the critical value jumps to 6.635. For 5 degrees of freedom, it’s 15.09. The stricter your threshold, the larger your chi-square value needs to be.
Large Samples Can Inflate Significance
One important caveat: the chi-square test is sensitive to sample size. With very large samples, even trivially small differences between groups can produce large chi-square values and small p-values. A study with 21,000 participants might find a statistically significant difference that has no practical importance whatsoever.
This is why researchers pair the chi-square test with an effect size measure, most commonly Cramér’s V. This statistic ranges from 0 to 1 and tells you how strong the relationship actually is, regardless of sample size. Using benchmarks developed by the statistician Jacob Cohen: a Cramér’s V between 0.05 and 0.15 is a small effect, 0.15 to 0.25 is medium, and above 0.25 is large. A significant chi-square with a Cramér’s V of 0.06 tells you the relationship is real but practically negligible.
Minimum Requirements for Valid Results
A chi-square result is only trustworthy if your data meets certain conditions. The most widely cited rule, known as Cochran’s rule, states that expected frequencies in each cell of your table should generally be at least 5. In practice, the rule is more flexible than many textbooks suggest. For tables with more than one degree of freedom, you can tolerate a minimum expected frequency of 1 as long as only a small proportion of cells fall below 5 (roughly 1 in 5 cells, or 2 in 10).
For 2×2 tables, the standards are stricter. If your total sample is under 20, or if it’s between 20 and 40 with any expected cell count below 5, Fisher’s exact test is the better choice. For 2×2 tables with a total sample above 40, the standard chi-square test works well. Some statistical software applies Yates’s continuity correction to 2×2 tables, which adjusts the chi-square value downward to account for the fact that you’re using a continuous distribution to approximate discrete (counted) data. This correction tends to be conservative, meaning it makes significance harder to achieve, and many statisticians prefer to skip it in favor of the uncorrected test.
How to Check Your Own Result
If you have a chi-square value and want to know whether it’s significant, follow these steps. First, determine your degrees of freedom from the structure of your data. Second, choose your significance level (0.05 unless you have a reason to use something else). Third, look up the critical value in a chi-square distribution table or use statistical software to get the p-value directly. If your chi-square exceeds the critical value, or if your p-value is below your threshold, the result is significant.
Then ask the more important question: is the effect large enough to matter? A significant p-value tells you the pattern in your data is unlikely to be noise. It doesn’t tell you the pattern is meaningful, important, or useful. Calculating Cramér’s V or another effect size measure gives you that second piece of the puzzle.