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The chi-squared (χ²) test is a statistical test used to determine whether there is a statistically significant difference between observed results and expected results. In genetics, it is used to test whether offspring ratios from a cross match the expected Mendelian ratios, or whether observed deviations are simply due to chance.
When performing a genetic cross, the observed results rarely match the expected ratio perfectly. For example, a monohybrid cross (Tt × Tt) is expected to produce a 3:1 ratio, but an actual experiment might yield 82 tall and 18 dwarf plants (instead of 75:25 from 100 offspring). The question is: is this deviation significant, or could it be due to random chance?
The chi-squared test provides a rigorous, mathematical way to answer this question.
Every chi-squared test begins with a null hypothesis (H₀):
The null hypothesis states that there is no significant difference between the observed and expected results. Any difference is due to chance alone.
χ² = Σ [(O − E)² / E]
Where:
A cross between two heterozygous pea plants (Tt × Tt) produces 100 offspring:
| Category | Observed (O) | Expected (E) | O − E | (O − E)² | (O − E)² / E |
|---|---|---|---|---|---|
| Tall | 82 | 75 | 7 | 49 | 0.653 |
| Dwarf | 18 | 25 | −7 | 49 | 1.960 |
| Total | 100 | 100 | χ² = 2.613 |
Degrees of freedom: df = 2 − 1 = 1
Critical value at p = 0.05 with 1 df: 3.841
Conclusion: χ² (2.613) < critical value (3.841), so we accept the null hypothesis. There is no significant difference between observed and expected results. The data are consistent with a 3:1 ratio.
A dihybrid cross (RrYy × RrYy) produces 640 offspring:
Calculating expected values: Total offspring = 640. Expected proportions: 9/16, 3/16, 3/16, 1/16.
| Category | Observed (O) | Expected (E) | O − E | (O − E)² | (O − E)² / E |
|---|---|---|---|---|---|
| Round yellow | 370 | 360 | 10 | 100 | 0.278 |
| Round green | 124 | 120 | 4 | 16 | 0.133 |
| Wrinkled yellow | 110 | 120 | −10 | 100 | 0.833 |
| Wrinkled green | 36 | 40 | −4 | 16 | 0.400 |
| Total | 640 | 640 | χ² = 1.644 |
Degrees of freedom: df = 4 − 1 = 3
Critical value at p = 0.05 with 3 df: 7.815
Conclusion: χ² (1.644) < critical value (7.815), so we accept the null hypothesis. The data are consistent with a 9:3:3:1 ratio, supporting independent assortment.
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