Chi-Square Test Calculator

This chi-square test calculator tells students, researchers and survey analysts whether counts in categories differ from what chance would produce, using a test of independence for tables or a goodness of fit test for a single list. It applies Pearson's chi-squared test exactly, shows expected counts and gives an exact p value. Formulas checked October 2026.

Quick answer: the chi-square statistic adds up (observed − expected)² ÷ expected across every cell. In the example loaded in this chi-square test calculator, 200 pupils' travel choices give χ² = 11.50 with 2 degrees of freedom and p = 0.0032, so travel mode and year group are linked at the 5% level.

What Is a Chi-Square Test?

A chi-square test is a hypothesis test for count data that measures how far observed frequencies sit from the frequencies expected under the null hypothesis. In words: for each cell, square the gap between observed and expected, divide by expected, then add the results. A large total means the data fit the null hypothesis badly.

The total is compared with a chi-square distribution on the right degrees of freedom. The calculator works out the upper tail area with the regularised incomplete gamma function, so the p value is exact rather than read from a printed table with only a few columns.

How to Use the Chi-Square Test Calculator

Step 1: Choose the Type of Test

Pick Independence when you have two categorical variables in a table, such as year group by travel mode. Pick Goodness of fit when you have one variable and a claimed distribution, such as a die that should land on each face equally often.

Step 2: Paste Your Observed Counts

Type or paste one row per line, with values separated by commas or tabs. Column names on the first line and a label at the start of each row are optional, and the chi-square test calculator detects them. For goodness of fit you can also enter custom expected shares such as 9, 3, 3, 1; the ratio calculator helps you simplify or scale a ratio like that before you paste it in.

Step 3: Set Yates Correction and Alpha

Tick Yates continuity correction only for a 2 × 2 table when you want a more cautious result. Then choose a significance level; 0.05 is standard in most coursework and journals.

Step 4: Read the Result

The chi-square test calculator gives you the p value, χ², degrees of freedom, the critical value and an effect size. The table lists each cell's observed and expected count and its share of χ², so the biggest numbers in the last column show where the difference comes from.

Which Chi-Square Test Do I Need?

Use a test of independence for a two way table of counts and a goodness of fit test for one list of counts against expected proportions. Both use the same formula in the chi-square test calculator; only the expected counts and degrees of freedom change.

Feature	Test of independence	Goodness of fit
Data	Two categorical variables in a table	One categorical variable
Null hypothesis	The variables are unrelated	Counts follow the stated proportions
Expected count	row total × column total ÷ grand total	total × expected proportion
Degrees of freedom	(rows − 1) × (columns − 1)	categories − 1
Effect size	Cramér's V	Cohen's w

How Do You Calculate a Chi-Square Test by Hand?

Work out each expected count, then add (O − E)² ÷ E across all cells and look up the p value for that total on (rows − 1) × (columns − 1) degrees of freedom. Here is the example loaded in the tool.

Owen, a geography teacher, surveys 100 Year 7 and 100 Year 11 pupils about how they get to school. Year 7: 38 walk, 12 cycle, 50 come by car. Year 11: 52 walk, 21 cycle, 27 by car. The walk column totals 90, so each expected walk count is 100 × 90 ÷ 200 = 45; cycling expects 16.5 and car 38.5 in each row.

Cell	Observed	Expected	(O − E)² ÷ E
Year 7, walk	38	45	1.089
Year 7, cycle	12	16.5	1.227
Year 7, car	50	38.5	3.435
Year 11, walk	52	45	1.089
Year 11, cycle	21	16.5	1.227
Year 11, car	27	38.5	3.435
Total χ²	200	200	11.502

With (2 − 1) × (3 − 1) = 2 degrees of freedom, the critical value at 0.05 is 5.991. Owen's χ² of 11.50 is well above it, and p = 0.0032. Cramér's V is 0.24, a small but real association, and the car cells contribute most: younger pupils are driven far more often.

What Does a Significant Chi-Square Result Mean?

A significant result, p below 0.05, means counts this far from expected would occur less than 5% of the time if the null hypothesis were true. It shows an association, not a cause, and it does not say how strong the link is; that is what Cramér's V or Cohen's w adds.

A non significant result is also informative. Switch to Goodness of fit and the loaded dice data (15, 22, 18, 27, 14 and 24 from 120 rolls) give χ² = 6.70 on 5 degrees of freedom and p = 0.244, so there is no evidence the die is biased. If your outcome is a conversion rate for two web pages, the A/B testing significance calculator runs the equivalent two proportion test and adds a sample size planner, which makes it the better choice for marketing experiments.

When Should You Use Yates' Correction?

Use Yates' correction only on 2 × 2 tables, mainly when expected counts are small, because it subtracts 0.5 from each |O − E| and makes the test more conservative. Take a clinic that sent appointment reminder texts: 30 of 50 patients who got a text attended, against 18 of 50 who did not. Without correction χ² = 5.77 and p = 0.0163; with Yates's correction χ² = 4.85 and p = 0.0277. Both are significant here, but near the 0.05 line the choice can change your conclusion, so report which one you used.

A chi-square test calculator works on counts, not means. If you measured something on a continuous scale, such as test scores or reaction times, the t-test calculator is the right tool for comparing two group averages, while this one handles categories.

Key Chi-Square Test Terms Explained

These definitions explain the statistics behind the chi-square test calculator, so you can read the p value, degrees of freedom and effect size correctly.

Null hypothesis

The default assumption a statistical test sets out to challenge, such as two categorical variables being unrelated or a die being fair. A small p value counts as evidence against it, not as proof that the alternative is true.

Expected frequency

The count each cell would hold if the null hypothesis were exactly true. In a contingency table it equals the row total multiplied by the column total, divided by the grand total of all observations.

Degrees of freedom

The number of values in a calculation that are free to vary once the totals are fixed. For a table it equals the number of rows minus one multiplied by the number of columns minus one.

P value

The probability of seeing a test statistic at least as extreme as the one observed if the null hypothesis held. Comparing it with the chosen significance level decides whether the result is reported as statistically significant.

Contingency table

A grid that cross tabulates counts for two categorical variables, with one variable in the rows and the other in the columns. It is the standard input for a chi-square test of independence.

Cramér's V

An effect size for chi-square tests that scales the statistic to a value between 0 and 1. It shows how strong an association is, which matters because very large samples can make tiny differences statistically significant.

Frequently Asked Questions