Math · Introductory statistics · Worked example
Test a 3:1 genetic ratio
A monohybrid cross is expected to give tall and short plants in a 3:1 ratio. Of 200 offspring, 140 are tall and 60 short. Test the 3:1 model at α = 0.05.
Find the expected counts
Three quarters of 200 is 150 tall, and one quarter is 50 short.
Compute χ²
Both categories are 10 away from expectation, but the smaller category contributes more.
Find the p-value
Two categories give 1 degree of freedom, and P(χ² ≥ 2.67) ≈ 0.10.
Decide
0.10 > 0.05, so the counts are consistent with a 3:1 ratio. The test assumes the offspring are independent and the cross is as described.
Result
χ² ≈ 2.67 with 1 degree of freedom and p ≈ 0.10: consistent with a 3:1 ratio.
Your turn
A dihybrid cross gives 90, 30, 32 and 8 offspring against a 9:3:3:1 expectation of 90, 30, 30 and 10. Find χ².
Show the answer and explanation
χ² ≈ 0.53 with 3 degrees of freedom, p ≈ 0.91: an excellent fit.
Only the last two categories differ: 4/30 + 4/10 ≈ 0.133 + 0.4.
Keep exploring
In Statistics, Distributions shows P(X ≥ 2.66667) ≈ 0.1025 with 1 degree of freedom. At 130 tall and 70 short, χ² would be 10.67 and p about 0.001.
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