Free Chi-squared Test OCR A Level Biology revision notes – covering specification point 6.1.2(c).
The chi-squared (χ²) test is a statistical test used to determine whether the difference between observed and expected results is statistically significant, or whether it is due to chance.
Calculating the chi-squared value from the observed results and the expected results allows for the difference between them to be determined as being either statistically significant or due to chance.
A statistically significant result has a high chance of being caused by an underlying mechanism (and not due to random chance), whilst a result that is not significant has a high chance of being due to random chance (and not due to an underlying mechanism).
The chi-squared test tests the null hypothesis, and accepts it if there is no significant difference, and rejects it if there is a significant difference. The alternative hypothesis is rejected in the former outcome and accepted in the latter outcome.
The Chi-Squared Formula
The value of chi-squared (χ²) is calculated using the formula:
χ² = ∑ (O − E)² / E
The formula is also shown like this:
Where:
- O = the observed value in each category
- E = the expected value in each category
- ∑ = the sum of
- Χ² = the chi-squared value
It is useful to know that the smaller the value of χ², the better the fit between the observed and expected data.
Worked Example
288 F₂ generation plants were grown from a dihybrid cross between two heterozygous individuals, investigating the inheritance of seed colour and seed shape.
The predicted ratio is 9:3:3:1, so the expected values are:
- 9/16 of 288 = 162 yellow and round
- 3/16 of 288 = 54 green and round
- 3/16 of 288 = 54 yellow and wrinkled
- 1/16 of 288 = 18 green and wrinkled
χ² is calculated using the formula:
χ² = ∑ (O − E)² / E
For convenience, the calculation is typically processed in the following format:
| Phenotype | Observed (O) | Expected (E) | O – E | (O – E)2 | (O – E)2 / E |
|---|---|---|---|---|---|
| Yellow, round | 169 | 162 | 7 | 49 | 0.30 |
| Green, round | 54 | 54 | 0 | 0 | 0.00 |
| Yellow, wrinkled | 51 | 54 | −3 | 9 | 0.17 |
| Green, wrinkled | 14 | 18 | −4 | 16 | 0.88 |
| Sum (Σ) | 1.35 | ||||
The calculated value of χ² is 1.35.
The degrees of freedom are required to read the correct critical value from a chi-squared probability table.
There are 4 phenotype categories, so the degrees of freedom are:
df = 4 − 1
df = 3.
A significance level of 0.05 is expected as standard unless otherwise stated.
In this case, the significance level at 3 degrees of freedom for p = 0.05 is 7.82.
Because 1.35 is less than the critical value of 7.82, the difference between observed and expected results is not statistically significant. The null hypothesis is accepted, and the difference between the observed counts and the predicted 9:3:3:1 ratio is attributed to chance. There is a greater than 5% probability that the difference is due to chance.