In a chi-squared goodness-of-fit test, keep fractional expected counts in the calculation. The observed counts are whole numbers because they came from counting individuals. Expected counts describe what a model predicts on average, so they can be fractions.
The 2027 H2 Biology syllabus (9477) includes applying a chi-squared test to results such as those from breeding experiments. This worked example turns a 9:3:3:1 model into four expected counts and shows how to write a conclusion without overstating what the test found.
- Use the observed total to scale the predicted ratio.
- Keep the expected values at full precision.
- Compare the statistic with the critical value for the stated test.
Worked example | Four offspring classes
An original exercise gives 60, 18, 17 and 5 offspring in four distinct phenotype classes. The null hypothesis is that the proportions follow a fixed 9:3:3:1 ratio. The total observed count is 100 and the ratio has 16 parts.
Each part of the ratio is 100 ÷ 16 = 6.25 expected offspring. The expected counts are 56.25, 18.75, 18.75 and 6.25. They sum to 100. Keeping the fractions preserves the stated model.
| Phenotype class | Observed | Expected |
|---|---|---|
| 1 | 60 | 56.25 |
| 2 | 18 | 18.75 |
| 3 | 17 | 18.75 |
| 4 | 5 | 6.25 |
| Total | 100 | 100.00 |
For the first class, (60 − 56.25)² ÷ 56.25 = 0.25. The next three contributions are 0.03, approximately 0.1633 and 0.25. The sum is approximately 0.6933.
Compare with the critical value
There are four classes and no model parameters are estimated from the data here, so there are 4 − 1 = 3 degrees of freedom. Suppose the question supplies a 5% critical value of 7.815. The statistic is below that value, so fail to reject the null hypothesis at the 5% significance level.
A suitable conclusion is “The observed counts do not differ significantly from the expected 9:3:3:1 proportions at the 5% level.” This tells the reader what the data support under this test. It leaves room for other models that might also fit the observations.
Check the expected column before using the formula
Rounding each expected count independently to a whole number can distort the proportions. In this example 56, 19, 19 and 6 happen to sum to 100, but the test contributions still change. Use the model’s counts, and round the reported test statistic only as requested.
For a fresh practice, suppose 84 offspring are expected to follow a 3:1 ratio. The expected counts are 63 and 21. If the total changes to 86, the expected counts are 64.5 and 21.5. Both columns reflect the same theoretical proportions.
Ask a study partner to check that the expected counts sum to the observed total and that the ratio used matches the null hypothesis. Those checks can catch a setup error before you square any differences.
Frequently asked questions
Can an observed count be 6.25?
An observed number of offspring is a whole-number count. An expected value can be fractional.
Does failing to reject prove the genetic model?
It means the test did not find a statistically significant departure at the chosen level, given the data and assumptions.
Should I calculate with very small expected counts?
Follow the question and your course guidance on whether the test assumptions are met. A small observed count and a small expected count are different issues.
Official information
SEAB’s 2027 A-Level H2 Biology syllabus (9477), Mathematical Requirements and Notes on the Use of Statistics in Biology. Checked 9 November 2026.