Goodness of fit: the chi-squared statistic

Hypotheses and statistic

• H0: the model fits (for example 'the die is fair', 'the data can be modelled by a Poisson distribution').
• H1: the model does not fit.

Compute expected frequencies E from the model (total × probability) and

X² = Σ (O - E)² / E, which is approximately χ² with ν degrees of freedom when H0 is true. An equivalent form is X² = Σ O²/E - N.

Hypotheses must name the model: examiners do not accept 'the claim is correct' on its own.

120 throws of a die

Degrees of freedom and critical values

ν = number of cells - 1 - number of parameters estimated from the data. The 1 is for the total, which the expected frequencies are forced to match.

The test is always upper-tailed: large X² means poor fit. Reject H0 if X² > the critical value χ²_ν(α) from the table.

Upper percentage points of χ² (Pearson tables)

Expected frequencies at least 5

The χ² approximation is poor if any expected frequency is less than 5. Then combine adjacent cells (adding O and E) before calculating X², and count cells after combining when finding ν. Yates' correction is not used in S3.

Example: a fair die

120 throws, E = 20 each. X² = (16 + 25 + 9 + 9 + 36 + 25)/20 = 6. ν = 6 - 1 = 5, critical value at 5% = 11.070. 6 < 11.070, so do not reject H0: there is no evidence that the die is biased.

Worked example

A shop believes sales of three sizes are in the ratio 3 : 2 : 1. In 240 sales the counts are 131, 70 and 39. Test at the 5% level.

1. H0: sales are in the ratio 3 : 2 : 1. H1: they are not.
2. E = 120, 80, 40.
3. X² = 11²/120 + 10²/80 + 1²/40 = 2.2833
4. ν = 3 - 1 = 2; critical value 5.991; 2.283 < 5.991.

Answer: Do not reject H0: the sales are consistent with the ratio 3 : 2 : 1.

Check yourself

1. A die is thrown 120 times with observed frequencies 16, 25, 17, 23, 14, 25 for scores 1 to 6. Calculate X² for the hypothesis that the die is fair.
   Show answer

   6. E = 20 each; Σ(O - E)² = 120; X² = 120/20 = 6.

2. In the die test, how many degrees of freedom are there?
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   5. 6 cells, minus 1 for the total, no parameters estimated: ν = 5.

3. Find the 5% critical value of χ² with 5 degrees of freedom. Give your answer to 3 decimal places.
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   11.070. From the table: 11.070.

4. Sales of three sizes are believed to be in the ratio 3 : 2 : 1. In 240 sales the counts are 131, 70 and 39. Calculate X². Give your answer to 3 significant figures.
   Show answer

   2.28. E = 120, 80, 40; X² = 2.2833.

5. Observed frequencies 30, 50, 20 have expected frequencies 25, 50, 25. Calculate X² using ΣO²/E - N.
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   2. 36 + 50 + 16 - 100 = 2.

6. The die test gives X² = 6 with critical value 11.070. Which conclusion is correct?
   • Reject H0: the die is biased
   • Do not reject H0: there is no evidence that the die is biased
   • Do not reject H0: the die is proved to be fair
   • Reject H0, because X² is positive
   Show answer

   Do not reject H0: there is no evidence that the die is biased. 6 < 11.070, so the fit is acceptable. A test never proves H0.

7. Expected frequencies in a goodness of fit test are 18.2, 12.6, 6.1, 2.4 and 0.7. What should you do before calculating X²?
   • Delete the cells with E < 5
   • Use Yates' correction
   • Multiply the small expected frequencies by 5
   • Combine the last three cells, so every expected frequency is at least 5
   Show answer

   Combine the last three cells, so every expected frequency is at least 5. 6.1 + 2.4 + 0.7 = 9.2 ≥ 5 (combining only the last two gives 3.1). The observed frequencies are combined in the same way.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) mark scheme, June 2024
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel Mathematical Formulae and Statistical Tables

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