# Chi-Square Test Calculator Run a Chi-Square goodness-of-fit test on observed vs. expected category counts and get the test statistic, degrees of freedom, and p-value. --- - **Canonical URL:** https://dothecalculation.com/calculators/chi-square-test-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Chi-Square Test Calculator Run a Chi-Square goodness-of-fit test on observed vs. expected category counts and get the test statistic, degrees of freedom, and P-value. - Goodness-of-fit testing on any number of categories - Automatic degrees of freedom (k − 1) - P-value and significance verdict ## Introduction to Chi-Square Hypothesis Testing The Chi-Square (\(\chi^2\)) test is a fundamental inferential statistics method used to analyze categorical data. Developed by Karl Pearson in 1900, it helps researchers determine whether there is a significant difference between expected categorical frequencies and observed frequencies. It is widely used in sociology, biology, marketing research, and medical trials to evaluate survey responses, genetic traits, and demographic variables. There are two main types of Chi-Square tests: the Goodness-of-Fit test and the Test of Independence. The Goodness-of-Fit test determines if a sample distribution matches a hypothesized population distribution (e.g., checking if a die is fair). The Test of Independence evaluates whether two categorical variables (such as gender and voting preference) are associated, using a contingency table. **This calculator implements the Goodness-of-Fit version only** — a single list of observed counts against a single list of expected counts. Enter your observed category counts and the expected counts you are testing against (equal-length comma-separated lists), and the solver computes the test statistic, degrees of freedom (\(df = k - 1\)), and the P-value using a Wilson-Hilferty normal approximation to the Chi-Square distribution. ## How to Use This Calculator Enter your observed category frequencies as a comma-separated list (e.g. "25, 45, 15, 15"), then enter the corresponding expected frequencies in the same order and of the same length. The calculator computes the Chi-Square statistic, degrees of freedom, and P-value, and flags whether the result is statistically significant at the 0.05 level. ## Worked Example: Observed [25, 45, 15, 15] vs. Expected [20, 40, 20, 20] $$\chi^2 = \frac{(25-20)^2}{20} + \frac{(45-40)^2}{40} + \frac{(15-20)^2}{20} + \frac{(15-20)^2}{20} = 1.25 + 0.625 + 1.25 + 1.25 = 4.375$$ $$df = k - 1 = 4 - 1 = 3, \qquad P(\chi^2 \geq 4.375) \approx 0.2223 \; (22.23\%)$$ Since the P-value (≈22.2%) is well above the common 0.05 significance threshold, this result would fail to reject the null hypothesis — the observed counts are not significantly different from what was expected. ## Chi-Square Formula and Degrees of Freedom The fundamental formula to compute the Chi-Square statistic is: $$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$ Where \(O_i\) is the observed frequency in category \(i\), and \(E_i\) is the expected frequency in category \(i\). The degrees of freedom (\(df\)) determine the shape of the Chi-Square distribution. For the Goodness-of-Fit test this calculator runs, \(df = k - 1\), where \(k\) is the number of categories. (A separate Test of Independence on a contingency table instead uses \(df = (r - 1)(c - 1)\), based on table rows \(r\) and columns \(c\) — that mode isn't implemented here, but the FAQ below covers the formula for reference.) ## Evaluating P-Values and Critical Boundaries Once the test statistic is computed, we compare it against critical values from the Chi-Square distribution or evaluate the P-value (the probability of obtaining a test statistic at least as extreme as the observed one, assuming the null hypothesis is true). If the P-value is less than the significance level \(\alpha\) (typically 0.05), we reject the null hypothesis, concluding that the difference is statistically significant. ## Related Calculators To test whether a sample mean differs significantly from a hypothesized value, use the [t-test calculator](/calculators/t-test-calculator). For a range of plausible values around a sample estimate rather than a significance verdict, see the [confidence interval calculator](/calculators/confidence-interval-calculator). ## Frequently asked questions ### What is a Chi-Square test? A Chi-Square test is a statistical hypothesis test used to analyze differences between observed and expected frequencies in categorical data. ### What is the Goodness-of-Fit test? It is a test that evaluates whether an observed sample distribution fits a theoretical distribution — the version this calculator computes. ### Does this calculator run a Test of Independence on a contingency table? No — this tool runs the Goodness-of-Fit test only (one list of observed counts vs. one list of expected counts). A Test of Independence compares two categorical variables in a contingency table and isn't available in this tool yet. ### How would expected values be calculated in an independence test? For each cell in a contingency table, the expected value is Row Total × Column Total / Grand Total. This is general chi-square background, not something this calculator's single-list mode computes. ### What is the Chi-Square formula? The formula is χ² = Σ (O − E)² / E, where O is observed and E is expected, summed across every category. ### What are degrees of freedom (df)? Degrees of freedom represent the number of values in the final calculation that are free to vary. For this calculator's goodness-of-fit test, df = k − 1, where k is the number of categories. ### What is a P-value? The P-value is the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true. ### What is a critical value? A critical value is a boundary point on the distribution curve compared against the test statistic to determine rejection of the null hypothesis. ### What is the null hypothesis for a goodness-of-fit test? The null hypothesis (H₀) states that the observed frequencies match the expected frequencies — that any difference is due to random sampling variation, not a real effect. ### What is the minimum expected count requirement? For the Chi-Square approximation to be valid, expected category counts should generally be 5 or greater. With smaller expected counts, the P-value approximation becomes less reliable. ### Can Chi-Square values be negative? No. Because the differences (O − E) are squared, the Chi-Square statistic is always positive or zero. ### How does this calculator compute the P-value? It uses the Wilson-Hilferty transformation, a normal approximation to the Chi-Square distribution, rather than looking up an exact value from a Chi-Square table — fast and accurate to about 3-4 decimal places for typical degrees of freedom. ## Related concepts - **Goodness-of-Fit Test** — A chi-square test checking whether observed category counts match a hypothesized distribution. - **P-Value** — The probability of a result at least as extreme as the observed one, assuming the null hypothesis is true. - **Null Hypothesis** — The default assumption that there is no relationship or difference — here, that observed counts match expected counts. ## Related guides - [Standard Deviation Guide: Measure Data Spread Clearly](https://dothecalculation.com/blog/math/standard-deviation-explained) — Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator. ## Related calculators - [Student's t-Test Calculator](https://dothecalculation.com/calculators/t-test-calculator) — Run a one-sample Student's t-test against a hypothesized mean and get the t-statistic, degrees of freedom, and two-tailed p-value instantly. - [Percentile & Percentile Rank Calculator](https://dothecalculation.com/calculators/percentile-calculator) — Find the percentile value of a dataset or compute the percentile rank of a specific data point instantly with this free tool. - [Average Calculator](https://dothecalculation.com/calculators/average-calculator) — Use our free average calculator to find the mean, weighted average, and GPA. Shows step-by-step work for any set of numbers. - [Binomial Distribution Probability Calculator](https://dothecalculation.com/calculators/binomial-distribution-calculator) — Compute binomial probability mass function, cumulative distribution, mean, and variance instantly with this free statistics calculator. - [Combination Calculator](https://dothecalculation.com/calculators/combination-calculator) — Calculate combinations (nCr) instantly with step-by-step work, with or without repetition. Perfect for lottery odds, poker hands, and team selection. - [Confidence Interval Calculator](https://dothecalculation.com/calculators/confidence-interval-calculator) — Calculate confidence intervals for population means and proportions, including margin of error, with this free statistics calculator. --- _This mathematical solver is designed for academic, engineering, and educational analysis. Rounding errors, numerical tolerances, or algorithm constraints might apply near poles, boundary conditions, or complex coordinate spaces. Always verify critical computations independently._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/chi-square-test-calculator). Quote freely with attribution and a link to this page._