Chi-Square Calculator

What the calculator does

A chi-square (Χ²) test asks whether the counts you observed differ from the counts you would expect if some simple hypothesis were true. This calculator runs the two most common versions:

• Independence / Homogeneity. You supply a table of counts, with groups in rows and categories in columns. The null hypothesis is that the row a person falls in tells you nothing about the column, or equivalently that every group has the same distribution across the categories.
• Goodness-of-fit. You supply observed counts and the frequencies a model predicts. The null hypothesis is that the data come from that model, for example a die being fair.

The output has three lines: the Χ² statistic, the p-value and the degrees of freedom. A fourth line, WARNING, appears when the sample is too small for the chi-square approximation to be trustworthy.

How to enter your data

• Put one group per line and one category per column. Separate columns with spaces, commas or tabs, so a block copied from a spreadsheet pastes straight in.
• For a test of independence use whole-number counts, never percentages or means. Every line needs the same number of values, and the table needs at least two rows and two columns.
• For goodness-of-fit use exactly two columns: observed counts first, expected frequencies second. The frequencies must add up to 1, with a tolerance of 0.001, and the expected count for each row is its frequency times the total of the observed counts.
• Use a dot as the decimal mark. Do not leave blank lines or add a leading space to a line, because that is read as a missing value.

Swapping rows and columns does not change the result. Anything that is not a number is read as zero.

The formula

Χ² = Σ (O − E)² ÷ ESum over every cell: O is the observed count and E the expected count.

For a table of counts the expected value of a cell assumes rows and columns are unrelated:

E = (row total × column total) ÷ grand total

degrees of freedom = (rows − 1) × (columns − 1)For goodness-of-fit with k categories the degrees of freedom are k − 1.

The p-value is the probability of a Χ² at least as large as yours when the null hypothesis is true, taken from the upper tail of the chi-square distribution with those degrees of freedom. You can convert a statistic you already have with the chi-square to p-value calculator.

Worked example: test of independence

In a trial, 100 patients received a new treatment and 100 received a standard one. Of the first group 45 improved and 55 did not; of the second group 30 improved and 70 did not. Arrange the counts like this, and enter 45 55 on the first line and 30 70 on the second:

Row totals are 100 and 100, column totals are 75 and 125, and the grand total is 200. Every row total is 100, so the expected counts are 100 × 75 ÷ 200 = 37.5 and 100 × 125 ÷ 200 = 62.5 in each row.

Χ² = 7.5²/37.5 + 7.5²/62.5 + 7.5²/37.5 + 7.5²/62.5 = 1.5 + 0.9 + 1.5 + 0.9 = 4.80

There is (2 − 1) × (2 − 1) = 1 degree of freedom and the p-value is 0.0285. This is below 0.05, so at the 5% significance level the data are inconsistent with the two treatments having the same improvement rate.

A p-value says whether an association is detectable, not how large it is. A common effect size for a table is Cramér's V = √(Χ² ÷ (N × (min(rows, columns) − 1))), here √(4.8 ÷ 200) ≈ 0.155, a small to moderate association.

Worked example: goodness-of-fit

A die is rolled 120 times and the faces 1 to 6 come up 25, 17, 15, 23, 24 and 16 times. If the die is fair every face should appear 120 ÷ 6 = 20 times.

Χ² = (25−20)²/20 + (17−20)²/20 + (15−20)²/20 + (23−20)²/20 + (24−20)²/20 + (16−20)²/20 = 100/20 = 5.00

With six categories there are 5 degrees of freedom and the p-value is about 0.416, far above 0.05, so there is no evidence that the die is unfair. To run it in the calculator choose Goodness-of-fit and enter each count next to 0.1667 (1/6 rounded). The six frequencies add up to 1.0002, inside the allowed tolerance, and the tool reports Χ² ≈ 5.00 and p ≈ 0.416.

Critical values

If your Χ² is larger than the critical value for your degrees of freedom, the result is significant at that level. These are the standard upper-tail values:

For other significance levels, the critical value calculator gives the cut-off directly.

Assumptions and warnings

The p-value comes from a smooth chi-square curve that only approximates the real distribution of the statistic, and the approximation is poor when counts are small. The usual guidance is that each cell of a 2 by 2 table should be at least 5 and, for larger tables, that no more than about a fifth of the cells should fall below 5.

• The tool shows More than 20% of cells with values less than 5 detected as soon as any observed count in a test of independence is below 5.
• If every observed count is at least 5 but some expected count is below 5, it shows Cell with expected frequency less than 5 detected.
• No warning is given for goodness-of-fit, so check the expected counts (frequency × total) yourself.

Observations must also be independent: the same person should not appear in several cells. When counts are small, an exact test such as Fisher's exact test is a better choice than chi-square.

Common mistakes

• Entering percentages. The test depends on sample size, so it needs the raw counts. Percentages from different group sizes give meaningless results.
• Frequencies that do not sum to 1. In goodness-of-fit, the second column holds proportions, not counts. Divide the model's counts by their total first.
• A zero row or column. A row or column that sums to zero makes the expected counts zero, which the calculator reports as an error. Remove it.
• Reading significance as importance. With a very large sample even a trivial difference gives a tiny p-value. Look at the size of the difference as well.

To check whether the traffic split of an A/B test matches the plan, the sample ratio mismatch calculator applies a chi-square goodness-of-fit check for you.