Chi-Square Calculator — Goodness of Fit Test (Free)

A chi-square calculator tests whether observed counts match expected counts. Add (O − E)²/E over every category, then compare the total with the chi-square distribution. For a die rolled 60 times showing 8, 9, 12, 11, 6 and 14, each expected count is 10, so χ² = 4.2 with 5 degrees of freedom and p = 0.521: no evidence the die is unfair.

Embed this calculator on your site

Free to use on class pages, blogs and course pages. Copy the code below and paste it into your page:

<iframe src="https://www.effortlessmath.com/math-topics/chi-square-calculator/?em_embed=1" width="100%" height="720" style="border:0;max-width:100%" title="Chi-Square Calculator — Goodness of Fit Test (Free) by Effortless Math" loading="lazy"></iframe><p><a href="https://www.effortlessmath.com/math-topics/chi-square-calculator/">Chi-Square Calculator — Goodness of Fit Test (Free)</a> by <a href="https://www.effortlessmath.com/">Effortless Math</a></p>Copy code

How to use the chi-square calculator

1. Pick the test. Goodness of fit checks one categorical variable against expected counts or proportions. Independence checks whether two categorical variables in a contingency table are related. p-value turns a χ² statistic and its degrees of freedom into a p-value.
2. Enter the observed counts. For goodness of fit, leave the expected box empty when every category should be equally likely, or type expected counts, proportions, percentages or a ratio such as 9:3:3:1 (as 9 3 3 1). For independence, type the table and use + Row and + Column to resize it up to 10 × 10.
3. Choose the significance level α (0.05 is the usual choice) and read χ², df, the p-value and the decision. Open Show steps for the full working.

The result also shows expected counts, each cell’s contribution, residuals, the effect size (Cohen’s w or Cramér’s V), a warning when expected counts are too small, the chi-square curve with the p-value shaded, and a table of critical values for your df.

The chi-square statistic

χ² = Σ (O − E)² / E, where O is an observed count and E the expected count for the same category or cell. A large χ² means the observed counts are far from what H₀ predicts. Always use counts, not percentages, for O.

• Goodness of fit: E = N × (expected proportion), and df = k − 1 for k categories.
• Independence: E = (row total × column total) / N, and df = (rows − 1)(columns − 1).

The p-value is the area of the chi-square curve to the right of your statistic. The calculator finds it with the regularized incomplete gamma function, p = Q(df/2, χ²/2), which is accurate far into the tail, so very small p-values are shown in scientific notation instead of being rounded to 0.

Worked examples

Goodness of fit: is the die fair?

Rolls of 1 to 6: 8, 9, 12, 11, 6, 14 (N = 60). If the die is fair, E = 60 / 6 = 10 for each face. Contributions: 0.4, 0.1, 0.4, 0.1, 1.6, 1.6, so χ² = 4.2 with df = 5. The p-value is 0.521, far above 0.05, and the critical value 11.070 is far above 4.2. Fail to reject H₀.

Goodness of fit with a ratio: Mendel’s peas

Mendel counted 315, 108, 101 and 32 peas (N = 556) where theory predicts 9:3:3:1. The expected counts are 556 × 9/16 = 312.75, then 104.25, 104.25 and 34.75. The statistic is χ² = 0.470 with df = 3, p = 0.925: the data fit the 9:3:3:1 ratio very well.

Test of independence: a 2 × 3 table

Two groups of 50 people choose among options A, B and C: group 1 picks 25, 15, 10 and group 2 picks 15, 25, 10. The column totals are 40, 40 and 20, so the expected counts are 20, 20 and 10 in each row. χ² = 1.25 + 1.25 + 0 + 1.25 + 1.25 + 0 = 5.0 with df = (2 − 1)(3 − 1) = 2, p = 0.082. At α = 0.05 there is not enough evidence of an association; Cramér’s V = √(5 / (100 × 1)) = 0.224.

A 2 × 2 table with and without Yates’ correction

For the table 42, 18 / 28, 32 (N = 120), Pearson’s χ² = 6.72 with df = 1 and p = 0.0095. With Yates’ continuity correction, χ² = 5.794 and p = 0.016. Both are significant at 0.05; the correction always makes a 2 × 2 test a little more cautious.

Chi-square critical values table

Reject H₀ when your χ² is larger than the value in your df row and α column.

Expected counts below 5 and effect size

The chi-square test is an approximation that works when expected counts are large enough. A common rule (Cochran’s) asks that no expected count is below 1 and no more than 20% are below 5. When the rule fails, combine small categories, collect more data, or for a 2 × 2 table use Fisher’s exact test. The calculator counts the low cells and shades them for you.

A p-value says whether a difference is likely to be real, not how big it is. With a large N even a tiny difference becomes significant, so report an effect size too. Cramér’s V = √(χ² / (N × min(r − 1, c − 1))) runs from 0 (no association) to 1. For a 2 × 2 table it equals |φ|, and 0.1, 0.3 and 0.5 are the usual cut-offs for small, medium and large. For goodness of fit, Cohen’s w = √(χ² / N) uses the same cut-offs.

Common mistakes

Checking homework with the calculator

Work out the expected counts and contributions by hand first. Then compare with the table in the result: if your χ² differs, the first cell that does not match shows where the slip happened. Most textbook answers round to 2 or 3 decimals, so tiny differences in the last digit are normal.

Chi-square test practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What does chi-square calculate?

It measures how far observed counts are from the counts you would expect under a hypothesis. The goodness-of-fit test compares one variable with a claimed distribution; the test of independence checks whether two categorical variables are related.

How do you calculate a p-value from chi-square?

Find the area to the right of the statistic under the chi-square curve with the right df. Use the p-value tab, a table (which only gives a range, such as 0.05 < p < 0.10), or a function such as CHISQ.DIST.RT in a spreadsheet. For χ² = 3.841 with df = 1, p = 0.05.

How do you do a chi-square goodness-of-fit test on a graphing calculator?

On a TI-84, put the observed counts in L1 and the expected counts in L2, then choose STAT, TESTS, χ²GOF-Test, enter df = k − 1 and calculate. The calculator here gives the same χ² and p-value and shows the working.

Is it chi-square or chi-squared?

Both are used. “Chi-square” is common in American textbooks and “chi-squared” in British and mathematical writing. The symbol is χ².

Is chi-square only for 2 × 2 tables?

No. The test of independence works for any r × c table (this calculator goes up to 10 × 10), and the goodness-of-fit test works for any number of categories. Only Yates’ correction is limited to 2 × 2 tables.

Related calculators

• P-value calculator: p-values for z, t and other tests
• Sample size calculator: how many responses a survey needs
• Probability calculator: basic and conditional probability
• Binomial probability calculator: exact tests for two categories
• Standard deviation calculator: spread of numerical data