The chi-squared test

When to use chi-squared

Use chi-squared when:

• the data are nominal (frequencies/counts in categories, e.g. held the door: yes or no);
• you are looking for a difference or association between groups;
• the design is independent (each person is counted in one cell only).

The expected frequency in each cell should not be too small (a common rule is at least 5).

Calculating χ²

1. Put the observed frequencies (O) in a contingency table with row and column totals.

2. For each cell: expected (E) = row total x column total / grand total.

3. For each cell work out (O - E)² / E.

4. Add them: χ² = Σ (O - E)² / E.

5. Degrees of freedom df = (rows - 1) x (columns - 1). A 2 x 2 table has df = 1.

6. Compare with the critical value. For chi-squared, the observed value must be equal to or greater than the critical value to be significant.

Extract from Pearson's chi-squared critical values table

Worked data

An observation records whether men and women hold a door open for the person behind them.

Worked example

Calculate χ² for the door-holding data and decide whether the difference is significant (two-tailed, p ≤ .05).

1. Expected: Men held = 40 x 48 / 80 = 24; Men did not = 16; Women held = 24; Women did not = 16.
2. (O - E)² / E: 1.500, 2.250, 1.500, 2.250.
3. χ² = 7.50; df = (2 - 1) x (2 - 1) = 1.
4. Critical value (df 1, two-tailed, p ≤ .05) = 3.84. 7.50 ≥ 3.84, so significant: reject the null hypothesis.

Answer: χ² = 7.50, significant at p ≤ .05: women held the door more often in this sample.

Check yourself

1. In a 2 x 2 table, a row total is 40, a column total is 36 and the grand total is 80. What is the expected frequency for that cell?
   Show answer

   18.0. 40 x 36 / 80 = 18.

2. Observed: Group A yes 24, no 16; Group B yes 12, no 28. Calculate χ² to 2 decimal places.
   Show answer

   7.27. Expected values: 18, 22, 18, 22. Sum of (O - E)² / E = 7.27.

3. With χ² = 7.27, df = 1, two-tailed, p ≤ .05 (critical value 3.84), what do you conclude?
   • The difference is not significant; keep the null hypothesis
   • Use Spearman's rank instead
   • The difference is significant; reject the null hypothesis
   • The df must be 2
   Show answer

   The difference is significant; reject the null hypothesis. 7.27 ≥ 3.84.

4. What are the degrees of freedom for a 3 x 2 contingency table?
   Show answer

   2. (3 - 1) x (2 - 1) = 2.

5. Observed: Group A yes 10, no 30; Group B yes 16, no 24. Calculate χ² to 2 decimal places.
   Show answer

   2.05. Expected values: 13, 27, 13, 27. χ² = 2.05.

6. χ² = 2.05, df = 1, two-tailed, p ≤ .05, critical value 3.84. Is it significant?
   • Yes, because 2.05 is greater than 1
   • No, because 2.05 is less than 3.84
   • Yes, because any chi-squared value is significant
   • No, because chi-squared needs ordinal data
   Show answer

   No, because 2.05 is less than 3.84. The observed value is smaller than the critical value.

7. Which data suit a chi-squared test?
   • Ratings of aggression out of 10 for the same children twice
   • Hours of sleep and hours of screen time
   • Reaction times in milliseconds for one group
   • Numbers of boys and girls who did or did not share a toy
   Show answer

   Numbers of boys and girls who did or did not share a toy. Nominal frequencies in independent groups.

8. Chi-squared requires data at which level of measurement?
   • Ordinal only
   • Interval only
   • Nominal
   • Ratio only
   Show answer

   Nominal. Frequencies in categories are nominal.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Psychology specification, Issue 5 (February 2021)
• Pearson IAL Psychology component guide: WPS02 (Issue 2, November 2024)
• Pearson WPS03/01 question paper, January 2022 (formulae and critical values tables)

Last updated 8 November 2026. Spotted a mistake? Tell us.