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Walk-through

How to Use This Calculator

4 steps

1

Choose your test

Goodness-of-fit checks whether one categorical variable matches an expected distribution — for example, is a die fair, or does a sample match a known population split? Test of independence checks whether two categorical variables are related — for example, does smoking status affect disease rate? Pick the tab that matches your question.

2

Enter your observed counts

For goodness-of-fit, enter a label and an observed count for each category, then choose equal or custom expected counts. For independence, fill in the contingency table's observed count for every row/column combination — add rows or columns if you have more than two categories per variable.

3

Set your significance level

Pick α — the threshold for calling a result statistically significant. α = 0.05 (5%) is the conventional default; use 0.01 for a stricter test or 0.10 for a more lenient one.

4

Read χ², df, and p-value

The Result tab shows the chi-square statistic, degrees of freedom, p-value, and the reject/keep-the-null decision. If p is less than α, the observed counts differ from what was expected by more than chance alone would explain.

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Reference

Formula & Methodology

4 formulas

Chi-square statistic

χ² = Σ(O − E)² / E

Sum, over every category (goodness-of-fit) or cell (independence), of the squared difference between the observed count O and the expected count E, divided by E. Larger deviations and smaller expected counts both push χ² higher.

Degrees of freedom — goodness-of-fit

df = k − 1

With k categories, df is one less than the category count — fixing the total observed count removes one degree of freedom.

Degrees of freedom — independence

df = (r − 1)(c − 1)

For an r-row by c-column contingency table, df is the product of (rows − 1) and (columns − 1) — fixing the row and column totals removes one degree of freedom from each dimension.

Expected count — independence

E_ij = (row total_i × column total_j) / grand total

Under the assumption that the two variables are independent, the expected count in row i, column j is proportional to that row's share and that column's share of the grand total.

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Glossary

Key Terms Explained

7 terms

Chi-square (χ²) A test statistic that measures how far a set of observed category counts deviates from the counts you'd expect under a stated hypothesis. Larger χ² means a bigger mismatch between observed and expected.

Observed vs. expected Observed (O) is the actual count you recorded in each category or cell. Expected (E) is the count that theory (goodness-of-fit) or independence (contingency table) predicts you should see if there's no real effect.

Goodness of fit A chi-square test that compares one categorical variable's observed distribution against a specific expected distribution, such as "equal across all categories" or a known historical split.

Independence Two categorical variables are independent if knowing one tells you nothing about the other. The chi-square test of independence checks whether observed joint counts in a contingency table are consistent with independence.

Contingency table A grid that cross-tabulates observed counts for two categorical variables — one variable per row, the other per column — used as the input to a test of independence.

Degrees of freedom The number of values in the calculation that are free to vary once the totals are fixed. It sets the shape of the chi-square distribution used to convert χ² into a p-value.

p-value The probability of seeing a chi-square statistic at least this large if the null hypothesis (fit or independence) were actually true. A small p-value is evidence against the null hypothesis.

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Scenarios

Real-World Examples

3 worked examples

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Testing a die for fairness

Goodness-of-fit

Rolls per face (expected) 10 each, 60 total Observed counts 5, 8, 9, 8, 10, 20

χ² = 13.4 with df = 5 gives p ≈ 0.0198. Since p is below α = 0.05, reject the null hypothesis that the die is fair — face 6 rolling 20 times instead of the expected 10 is driving most of the deviation.

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Smoking status and disease

Test of independence (2×2)

Smoker: disease / no disease 20 / 15 Non-smoker: disease / no disease 10 / 25

Row and column totals give expected counts of 15, 20, 15, and 20. χ² = 5.83 with df = 1 gives p ≈ 0.0157 — below α = 0.05, so reject independence: in this sample, smoking status and disease status are related.

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Understanding degrees of freedom

A three-way preference survey

Categories 3 (Coffee, Tea, Neither) Degrees of freedom k − 1 = 2

With 3 categories in a goodness-of-fit test, df = 3 − 1 = 2 — only 2 of the 3 category counts are free to vary once the total sample size is fixed, since the third is determined by subtraction.

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Deep Dive

Understanding the Chi-Square Calculator

3 chapters · 4 min read

The chi-square test answers a simple question: are the counts you observed in a set of categories close enough to what you'd expect — under a stated hypothesis — that the difference could plausibly be chance? This calculator runs the two most common versions, goodness-of-fit and test of independence, and reports the chi-square statistic, degrees of freedom, and p-value so you can make that call.

How the chi-square test works

Every chi-square test compares an observed count O to an expected count E in each category or cell, using χ² = Σ(O − E)² / E. Squaring the difference means both over- and under-counts push χ² up, and dividing by E means the same absolute gap matters more in a category where you expected few observations than in one where you expected many. The resulting χ² is compared against the chi-square distribution for the appropriate degrees of freedom to get a p-value — the probability of seeing a deviation this large (or larger) if the null hypothesis were true.

Goodness-of-fit vs. independence — which do I need?

Use goodness-of-fit when you have one categorical variable and want to know if its distribution matches a specific expected pattern — a fair die, a known population split, or a claimed ratio. Use the test of independence when you have two categorical variables recorded together for the same subjects (a contingency table) and want to know if they're related — does treatment group affect outcome, does region affect preference. The math is the same formula in both cases; the only difference is how the expected counts are derived and how degrees of freedom is computed.

Reading the p-value and the low-expected-count guard

Compare the p-value to your chosen significance level α (0.05 by default): p < α means reject the null hypothesis — the observed pattern is unlikely to be chance alone. p ≥ α means you don't have enough evidence to rule out chance. The chi-square approximation itself relies on expected counts being reasonably large — as a rule of thumb, every cell should have an expected count of at least about 5. This calculator flags a warning whenever any expected count falls below that threshold, since the p-value becomes less reliable with very sparse categories or cells.

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Questions

Frequently Asked Questions

6 questions

What's the chi-square formula?+

χ² = Σ(O − E)² / E — sum the squared difference between each observed count O and its expected count E, divided by E, across every category or cell.

How do I find degrees of freedom for a goodness-of-fit test?+

df = categories − 1. With k categories, one degree of freedom is used up by fixing the total observed count.

How do I find degrees of freedom for a test of independence?+

df = (rows − 1) × (columns − 1). For an r × c contingency table, fixing the row and column totals removes one degree of freedom from each dimension.

How is the expected count calculated?+

For goodness-of-fit with equal expected counts, E = total observed ÷ number of categories (or your own custom values). For a test of independence, E for a cell is (its row total × its column total) ÷ the grand total.

What does p < 0.05 mean?+

It's statistically significant at the conventional threshold — the observed counts deviate from expected by more than random chance would typically produce, so you reject the null hypothesis (fit or independence).

What's the minimum expected count I should use?+

About 5 per category or cell. Below that, the chi-square distribution is a poor approximation and the p-value becomes unreliable — this calculator warns you when any expected count drops under 5.

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Chi-Square Calculator

Test settings

Goodness-of-fit data

Contingency table

Cell-by-cell breakdown

How to Use This Calculator

Choose your test

Enter your observed counts

Set your significance level

Read χ², df, and p-value

Formula & Methodology

Key Terms Explained

Real-World Examples

Testing a die for fairness

Smoking status and disease

Understanding degrees of freedom

Understanding the Chi-Square Calculator

How the chi-square test works

Goodness-of-fit vs. independence — which do I need?

Reading the p-value and the low-expected-count guard

Frequently Asked Questions

Get a branded PDF of your results

Chi-Square Calculator

Test settings

Goodness-of-fit data

Contingency table

Cell-by-cell breakdown

How to Use This Calculator

Choose your test

Enter your observed counts

Set your significance level

Read χ², df, and p-value

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Chi-Square Calculator

How the chi-square test works

Goodness-of-fit vs. independence — which do I need?

Reading the p-value and the low-expected-count guard

Frequently Asked Questions

Get a branded PDF of your results

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