There is no single “good” chi-squared value. Whether your result is meaningful depends on two things: how many degrees of freedom your test has and what significance level you’re using. A chi-squared value of 4.0 might be statistically significant in one test and completely unremarkable in another. The number only means something when you compare it to the right critical value for your specific setup.
Why the Number Alone Means Nothing
A chi-squared test measures how far your observed data falls from what you’d expect if nothing interesting were happening (the “null hypothesis”). A value of zero means your data perfectly matches expectations. The further your value climbs above zero, the more your data deviates from what chance alone would produce.
But how high is high enough? That depends on your degrees of freedom, which reflect the size and complexity of your data. For a simple 2×2 table comparing two groups on a yes/no outcome, you have 1 degree of freedom. The formula for a contingency table is (rows minus 1) multiplied by (columns minus 1). So a 3×4 table has 6 degrees of freedom. A goodness-of-fit test with five categories has 4. More degrees of freedom shift the entire distribution to the right, meaning you need a larger chi-squared value to reach significance.
Critical Values at Common Significance Levels
Most researchers use a significance threshold of 0.05, meaning there’s a 5% chance the result occurred by random chance alone. If your chi-squared value exceeds the critical value for your degrees of freedom at that threshold, the result is considered statistically significant. Here are the critical values from the National Institute of Standards and Technology:
- 1 degree of freedom: 3.841 (at 0.05) or 6.635 (at 0.01)
- 2 degrees of freedom: 5.991 or 9.210
- 3 degrees of freedom: 7.815 or 11.345
- 4 degrees of freedom: 9.488 or 13.277
- 5 degrees of freedom: 11.070 or 15.086
- 6 degrees of freedom: 12.592 or 16.812
- 7 degrees of freedom: 14.067 or 18.475
- 8 degrees of freedom: 15.507 or 20.090
- 9 degrees of freedom: 16.919 or 21.666
- 10 degrees of freedom: 18.307 or 23.209
The first number in each pair is the threshold for p < 0.05. The second is for p < 0.01, a stricter standard sometimes used when you want stronger evidence. If your calculated chi-squared value is higher than the critical value, your result is statistically significant at that level.
A Quick Example
Say you’re testing whether men and women prefer different brands of coffee using a 2×3 table (two genders, three brands). That gives you (2-1) times (3-1) = 2 degrees of freedom. You run your chi-squared test and get a value of 7.2. Looking at the table, the critical value at 2 degrees of freedom and the 0.05 level is 5.991. Since 7.2 is larger than 5.991, your result is significant: the preference pattern between men and women is unlikely to be due to chance.
If your value had come out to 4.5 instead, it would fall below 5.991, and you’d conclude there’s no statistically significant difference in coffee preference between the groups.
Two Types of Chi-Squared Tests
The interpretation works the same way for both major types, but they answer different questions. A goodness-of-fit test checks whether one set of observed data matches an expected distribution. For instance, you might test whether customer visits are evenly spread across weekdays. An independence test checks whether two variables in a contingency table are related, like whether smoking status is associated with lung disease. In both cases, a larger chi-squared value means a bigger gap between what you observed and what the null hypothesis predicted.
Significant Doesn’t Always Mean Important
Here’s where many people get tripped up. A statistically significant chi-squared value tells you that a relationship or difference exists, but it says nothing about how strong that relationship is. With a large enough sample, even trivially small differences can produce huge chi-squared values and tiny p-values. Research on this sensitivity has shown that sample sizes in the tens of thousands can inflate chi-squared values dramatically, making practically meaningless patterns appear highly significant.
To measure the actual strength of the relationship, you need an effect size metric like Cramér’s V. This value ranges from 0 to 1, where 0 means no association at all and 1 means a perfect relationship. As a rough guide, values around 0.1 suggest a weak association, 0.3 a moderate one, and 0.5 or above a strong one. If your chi-squared test is significant but Cramér’s V is 0.04, you’ve found a real pattern that’s too small to matter in practice.
When the Test Can Mislead You
The chi-squared test relies on having enough data in each cell of your table. When expected counts in any cell are very low, the math behind the test breaks down and results become unreliable. The standard rule is that expected frequencies should be at least 5 in each cell. If your data doesn’t meet this threshold, Fisher’s exact test is a more reliable alternative.
Small sample sizes can also work against you in the opposite direction. With too few observations, even a genuinely meaningful difference between groups may not produce a chi-squared value large enough to cross the significance threshold. You’d miss a real effect simply because you didn’t have enough data to detect it. This is why planning your sample size before collecting data matters: you want enough observations to give the test a fair shot at finding a real difference if one exists, but not so many that trivial differences get flagged as significant.