Quick answer
The chi-square statistic is χ² = Σ (O − E)² ÷ E with df = k − 1. Observed counts 20, 30, 50 against expected 25, 25, 50 give χ² = 2, 2 degrees of freedom and p = 0.3679.
Enter observed and expected counts. The calculator returns χ², the degrees of freedom and the p-value.
Related tools: T Statistic Calculator: t statistic from summary statistics. P-Value Calculator: P-value from a z, t, chi-square or F statistic. Critical Value Calculator: Critical values for z, t, chi-square and F distributions.
How to use the Chi-Square Statistic Calculator
- Type observed counts, for example 20, 30, 50.
- Type expected counts, for example 25, 25, 50.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Observed counts | Text, such as 20, 30, 50 | — |
| Expected counts | Text, such as 25, 25, 50 | — |
Formula and method
Formula
χ² = Σ (O − E)² ÷ E, df = k − 1
IEEE-754 double precision; shown to at most 6 decimal places.
Expected counts must be positive; the p-value is the right tail.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Observed counts = 20,30,50; Expected counts = 25,25,50.
| Result | Value |
|---|---|
| Chi-square | 2 |
| Degrees of freedom | 2 |
| p-value | 0.367879 |
The first line is the headline result (chi-square). Below it the chi-Square Statistic Calculator lists degrees of freedom and p-value, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
(20 − 25)²/25 = 1, (30 − 25)²/25 = 1 and (50 − 50)²/50 = 0, so χ² = 2. With p = 0.37, the data are consistent with the expected counts.
Conditions
Each expected count should be at least 5 for the approximation to hold. The lists must have equal length and expected values must be above zero. It tests goodness of fit; independence in tables needs expected counts computed from totals.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Expected counts: entering "1" shows "Enter at least 2 numbers." beside that field and no result.
Choosing between the Chi-Square Statistic Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [T Statistic Calculator](/statistics/t-statistic/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [P-Value Calculator](/statistics/p-value/) | test statistic, statistic value and test type | p-value |
| [Critical Value Calculator](/statistics/critical-value/) | distribution, significance level and tails | Critical value |
| [Degrees of Freedom Calculator](/statistics/degrees-of-freedom/) | test and n₁ | Degrees of freedom |
| [One-Sample T-Test Calculator](/statistics/t-test-calculator/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- Equal-length lists
- Expected > 0
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for expected counts to confirm this.
Frequently asked questions
What does a small p-value mean?
The data differ from what was expected.
What is df here?
Categories minus one.
Can I use proportions?
Use counts, not proportions: the statistic Σ(O − E)²/E depends on the sample size, which proportions would remove.
What does the Chi-Square Statistic Calculator calculate?
Goodness-of-fit chi-square from observed and expected counts. It returns Chi-square, Degrees of freedom and p-value.
What formula does the Chi-Square Statistic Calculator use?
χ² = Σ (O − E)² ÷ E, df = k − 1. Expected counts must be positive; the p-value is the right tail.
What do I need to enter in the chi-Square Statistic Calculator?
You enter Observed counts and Expected counts.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Equal-length lists.
What are the limits of this calculator?
Equal-length lists. Expected > 0.
What is an example of a chi-Square Statistic result?
With observed counts 20,30,50 and expected counts 25,25,50, the chi-square is 2.
Related calculators
- T Statistic Calculator — t statistic from summary statistics.
- P-Value Calculator — P-value from a z, t, chi-square or F statistic.
- Critical Value Calculator — Critical values for z, t, chi-square and F distributions.
- Degrees of Freedom Calculator — Degrees of freedom for one- and two-sample t-tests.
- One-Sample T-Test Calculator — One-sample t-test from summary statistics.
- One-Sample Z-Test Calculator — Z-test for a mean with known population standard deviation.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.