A chi-square table is a reference grid that tells you how large your calculated chi-square statistic needs to be before you can call a result statistically significant. The rows represent degrees of freedom, the columns represent significance levels (like 0.05 or 0.01), and the number where they intersect is your critical value. If your calculated statistic is larger than that critical value, the result is significant.
What the Rows and Columns Mean
The left-most column of a chi-square table lists degrees of freedom, often abbreviated as “df.” This number reflects how much your data is free to vary, and it depends on the size of your data table. For a test of independence (the most common type), the formula is: (number of rows minus 1) times (number of columns minus 1). A simple 2×2 table has 1 degree of freedom. A 3×3 table has 4. A 4×5 table has 12.
The top row lists probability values. This is where it gets slightly tricky, because different textbooks format their tables differently. Most standard chi-square tables show upper-tail probabilities, meaning the columns are labeled with values like 0.10, 0.05, 0.025, 0.01, and 0.001. These represent the probability of getting a chi-square value that large (or larger) by pure chance. The 0.05 column is the one you’ll use most often, since a 5% significance level is standard in most fields.
The numbers filling the body of the table are critical values. Each one is a threshold. For example, at 1 degree of freedom and a significance level of 0.05, the critical value is 3.841. That single number is probably the most commonly referenced value in the entire table.
How to Find Your Critical Value
Reading the table is a three-step process:
- Step 1: Calculate your degrees of freedom from your data. If you’re comparing two groups across three categories, your table is 2 rows by 3 columns, giving you (2-1) x (3-1) = 2 degrees of freedom.
- Step 2: Choose your significance level. In most coursework and published research, this is 0.05. Some fields or stricter analyses use 0.01.
- Step 3: Find the row for your degrees of freedom, move across to the column for your significance level, and read the number at the intersection. That’s your critical value.
A Worked Example
Say you ran a chi-square test comparing patient distributions across two hospital units and three social classes. Your data table has 2 rows and 3 columns, so degrees of freedom = (2-1) x (3-1) = 2. You’re using a 0.05 significance level. Looking at the chi-square table, you find the row for df = 2 and the column for 0.05. The critical value there is 5.991.
Now suppose your calculated chi-square statistic came out to 7.34. Since 7.34 is greater than 5.991, you reject the null hypothesis. The difference in patient distribution between the two units is statistically significant at the 0.05 level. If your statistic had been 4.10, it would fall below the threshold, and you would not reject the null hypothesis.
The Decision Rule
The rule is straightforward: if your calculated chi-square value is greater than the critical value from the table, the result is statistically significant. If it’s equal to or smaller than the critical value, it’s not. Chi-square tests are almost always upper-tail tests, meaning you’re only looking at whether your value lands in the extreme right end of the distribution. You don’t need to worry about two-tailed vs. one-tailed the way you might with a t-test.
A larger chi-square statistic means a bigger gap between what you observed and what you’d expect if nothing interesting were happening. The table simply tells you how big that gap needs to be before it’s unlikely to be caused by random chance alone.
Watch for Table Format Differences
Some tables label their columns as the area to the right of the critical value (the p-value itself, like 0.05). Others label columns as the area to the left (like 0.95, meaning 95% of the distribution falls below that value). Both give you the same critical values, but the column headers look different. If you see columns labeled 0.90, 0.95, 0.975, 0.99, and 0.999, you’re looking at a table formatted as “probability less than the critical value.” In that case, a 0.05 significance level corresponds to the 0.95 column, not a column labeled 0.05. The NIST reference tables use this format. Always check the table’s header or footnote to see which convention it follows before you start reading values.
Choosing a Significance Level
The 0.05 level (a 5% chance of a false positive) is the default in social sciences, biology, and most health research. The 0.01 level is more conservative and used when you want stronger evidence, such as in clinical trials or when testing multiple comparisons. The 0.001 level is reserved for situations where you need very high confidence. Each stricter level raises the critical value, making it harder to reach significance. At 1 degree of freedom, for instance, the critical value jumps from 3.841 at the 0.05 level to 6.635 at 0.01 and all the way to 10.828 at 0.001.
When the Chi-Square Table Doesn’t Apply
The chi-square test has a key assumption: every cell in your expected frequency table should have a value of at least 5. If any expected cell count drops below 5, the chi-square approximation becomes unreliable, and the critical value you pulled from the table may lead you to the wrong conclusion. In that situation, Fisher’s exact test is the appropriate alternative. This comes up most often with small sample sizes or when one category in your data is rare.
For 2×2 tables with small samples, some statisticians also recommend applying Yates’ continuity correction, which slightly adjusts the chi-square calculation downward to reduce the chance of a false positive. Your textbook or software may handle this automatically, but it’s worth knowing it exists so you’re not confused when your hand calculation doesn’t match a computer’s output.
Reporting Your Results
If you’re writing up results for a paper or assignment, the standard format includes the chi-square symbol, degrees of freedom in parentheses, the calculated value, and the p-value. According to APA guidelines, report the chi-square statistic to two decimal places and give the exact p-value to two or three decimals. If the p-value is extremely small, report it as p < .001 rather than writing out a long string of zeros. Degrees of freedom are written without commas, even for large numbers. You don’t need to define common statistical abbreviations like df or p in your text.