D3.2.21 (HL)—Chi-squared test

The chi-squared test compares observed and expected genetic results to judge whether differences could be due to chance in inheritance problems.

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Latest appearanceNovember 2021

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November 2021Paper1 ["HL"] · TZ035[ 1 ]D3.2.21 (HL)—Chi-squared test

November 2019Paper2 ["HL"] · TZ02(c)[ 2 ]D3.2.21 (HL)—Chi-squared test

Coverage 2019–2021 · Updated 16 Jul 2026

Statement	Meaning in a dihybrid goodness-of-fit test
null hypothesis, H₀	observed departures from the expected genetic ratio are due to chance sampling
alternative hypothesis, H₁	the departures are too large for that chance model; another explanation is needed

Convert the ratio to expected counts before calculating χ². For a total of 556 offspring and a 9:3:3:1 model: multiply 556 by 9/16, 3/16, 3/16 and 1/16 to obtain 312.75, 104.25, 104.25 and 34.75.

Categories must be mutually exclusive, observations should be independent and expected counts should be large enough for the approximation. The expected ratio must come from a biological model chosen before inspecting the deviations.

Chi-squared tests whether counts fit a specified expectation; it does not identify the true alternative mechanism when they do not fit.

$\chi^2=\sum \frac{(O-E)^2}{E}$

Category	O	E	(O − E)² / E
both dominant	315	312.75	0.016
A dominant only	108	104.25	0.135
B dominant only	101	104.25	0.101
both recessive	32	34.75	0.218
sum	556	556	χ² = 0.470

For four mutually exclusive categories with one fixed total and no parameters estimated from the data, degrees of freedom = categories − 1 = 3.

Use counts, not percentages, inside the formula. Keep expected values unrounded during calculation, and include every category contribution before comparing with a critical value.

A chi-squared table compares observed and expected Drosophila phenotype counts, sums category contributions to 0.47 and compares the result with the p equals 0.05 critical value for three degrees of freedom.

At p = 0.05	Statistical decision	Genetic conclusion
χ² ≤ critical value	fail to reject H₀	deviations are not significant; data are consistent with the expected ratio
χ² > critical value	reject H₀	deviations are significant; the stated ratio does not adequately explain the data

For df = 3, the p = 0.05 critical value is 7.815. Because 0.470 < 7.815, fail to reject H₀: the observed Drosophila counts are consistent with a 9:3:3:1 expectation.

Do not say the test proves H₀ or proves independent assortment. A non-significant result means the observed departure is small enough to be explained by sampling under the model.

A significant result directs further investigation. Possible causes include linkage, differential survival, non-random fertilization, scoring error or an incorrect dominance model; chi-squared alone does not choose among them.

meiosis predicts gamete combinations → chromosome loci determine whether genes assort independently or travel together → crossing over creates reciprocal recombinants → test-cross counts reveal parental phase and recombinant frequency → chi-squared tests whether departures from a chosen ratio exceed chance expectation

Offspring pattern	First model to examine
9:3:3:1 F2	two unlinked genes with complete dominance
1:1:1:1 test cross	four equally frequent gametes from an unlinked double heterozygote
two large and two small reciprocal test-cross classes	linked loci with crossing over
significant χ² departure	assumptions, linkage, viability, fertilization or scoring require review

Keep prediction and evidence separate: define the genetic model first, calculate expected counts from it, compare observations with the expectation, and change the model only when chromosome location or statistical evidence requires it.

[ 2 ]

The chi-squared value was calculated as shown. Deduce, with reasons, whether the observed ratio differed significantly from the expected Mendelian ratio.

c^{2}=\Sigma \frac{(\text { Observed }- \text { Expected })^{2}}{\text { Expected }}=1002.6

	Probability
Degrees of freedom	0.995	0.975	0.20	0.10	0.05	0.025	0.02	0.01	0.005	0.002	0.001
1	0.00004	0.001	1.642	2.706	3.841	5.024	5.412	6.635	7.879	9.550	10.828
2	0.010	0.051	3.219	4.605	5.991	7.378	7.824	9.210	10.597	12.429	13.816
3	0.072	0.216	4.642	6.251	7.815	9.348	9.837	11.345	12.838	14.796	16.266
4	0.207	0.484	5.989	7.779	9.488	11.143	11.668	13.277	14.860	16.924	18.467
5	0.412	0.831	7.289	9.236	11.070	12.833	13.388	15.086	16.750	18.907	20.515
6	0.676	1.237	8.558	10.645	12.592	14.449	15.033	16.812	18.548	20.791	22.458
7	0.989	1.690	9.803	12.017	14.067	16.013	16.622	18.475	20.278	22.601	24.322

Concept essentials

Chi-squared compares observed counts with expected counts.
Expected values must come from a stated genetic hypothesis or ratio.
A larger chi-squared value means a poorer fit to the expected ratio.

D3.2.21 (HL)—Chi-squared test

Exam analysis

Recent exam appearances

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Compare Chi-Squared with the Critical Value, Then State Only What the Test Supports

Inheritance Models Are Built from Chromosomes and Tested with Offspring Counts

Concept essentials