Real crosses never give perfect ratios. A litter of 160 peas will not split exactly 90 : 30 : 30 : 10. The chi-squared test tells you whether the gap between what you observed and what you expected is small enough to put down to chance.
The chi-squared (χ2) test compares observed (O) numbers in each category with the expected (E) numbers predicted by a genetic ratio. Start with a null hypothesis: there is no significant difference between observed and expected results, and any difference is due to chance. Calculate E from the total and the ratio, then χ2 = Σ (O − E)2 / E (the formula is provided). Find the degrees of freedom = number of categories − 1, and compare χ2 with the critical value at p = 0.05 from the table. If χ2 is less than the critical value, the difference is not significant and the null hypothesis is accepted; if it is greater, the difference is significant (probability less than 0.05 of arising by chance), so the null hypothesis is rejected and something else, such as linkage or epistasis, may be operating.
Fig. 1: a worked chi-squared test for a dihybrid test cross
Section 01
Observed against expected.
Press Grow 160 offspring several times. Each batch is random, so the observed numbers (yellow bars) wobble around the expected 90 : 30 : 30 : 10 (blue bars), and χ2 changes each time. Then tick Genes linked and grow again.
Null hypothesis: there is no significant difference between the observed and expected numbers; any difference is due to chance (for a dihybrid cross: the genes assort independently and the ratio is 9 : 3 : 3 : 1).
Expected numbers: share the total in the expected ratio. For 160 offspring and 9 : 3 : 3 : 1, E = 160 × 9/16 = 90, then 30, 30 and 10.
The statistic: for each category work out (O − E)2 / E, then add them: χ2 = Σ (O − E)2 / E. Squaring removes the sign; dividing by E scales each difference to the size of the class.
Big or small: the further the results are from the prediction, the larger χ2 becomes.
Examiner trap
Always use the actual numbers of offspring for O and E, never percentages or ratios. And E does not have to be a whole number: 100 × 3/4 = 75, but 50 × 3/4 = 37.5 is fine.
Stage 1 · Learn
Check what the sim just showed you
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In the simulator, how are the expected numbers for 160 offspring and a 9 : 3 : 3 : 1 ratio found?
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Why does the simulator use 3 degrees of freedom?
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A batch gives χ2 = 2.4 with no linkage. What does the simulator conclude?
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When Genes linked is ticked, χ2 is usually very large. Why?
Section 02
Degrees of freedom and the critical value.
The simulator uses 3 degrees of freedom and a critical value of 7.82. Here is where those numbers come from, and how to word the conclusion.
Degrees of freedom
1
2
3
4
critical value at p = 0.05
3.84
5.99
7.82
9.49
Degrees of freedom = number of categories (phenotype classes) − 1. A monohybrid 3 : 1 cross has 1; a 9 : 3 : 3 : 1 cross has 3.
p = 0.05: biologists accept a difference as significant if it would arise by chance less than 5% of the time.
χ2 less than the critical value: p > 0.05; the difference is not significant, it is probably due to chance, so accept the null hypothesis: the results fit the expected ratio.
χ2 greater than the critical value: p < 0.05; the difference is significant, so reject the null hypothesis. Suggest a reason, for example that the genes are linked, show epistasis, or one genotype has lower survival.
Examiner trap
A non-significant result does not prove the hypothesis is true; it means the data are consistent with it. Say "the difference is not significant" rather than "there is no difference".
Stage 2 · Exam
Exam-style questions
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In a monohybrid cross, 120 offspring are expected in a 3 : 1 ratio. 82 show the dominant and 38 the recessive phenotype. What is χ2?
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Using the table of critical values, which conclusion is correct for χ2 = 6.20 with 2 degrees of freedom?
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A dihybrid cross gives χ2 = 25.3 against an expected 9 : 3 : 3 : 1 ratio. Which is the most likely explanation?
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Which is the correct null hypothesis for a chi-squared test on a test cross expected to give 1 : 1 : 1 : 1?
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