The Chi-Squared Test

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.

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Why Do We Need the Chi-Squared Test?

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.

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The Null Hypothesis

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.

• If the chi-squared test shows that the deviation is not significant, we accept (fail to reject) the null hypothesis — the results are consistent with the expected ratio.
• If the deviation is significant, we reject the null hypothesis — the results do not fit the expected ratio, and some other factor (e.g., linkage, epistasis, selection) must be responsible.

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The Chi-Squared Formula

χ² = Σ [(O − E)² / E]

Where:

• O = observed number in each category
• E = expected number in each category (calculated from the expected ratio and total sample size)
• Σ = sum across all categories

Steps for Calculating Chi-Squared

1. State the null hypothesis — e.g., "There is no significant difference between observed and expected 3:1 ratio."
2. Calculate expected values — multiply the expected proportion by the total number of offspring.
3. Calculate (O − E)² / E for each category.
4. Sum all values to obtain χ².
5. Determine degrees of freedom — df = number of categories − 1.
6. Compare χ² to the critical value from the chi-squared table at the chosen significance level (usually p = 0.05).
7. Draw a conclusion — accept or reject the null hypothesis.

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Worked Example 1 — Monohybrid Cross

A cross between two heterozygous pea plants (Tt × Tt) produces 100 offspring:

• Observed: 82 tall, 18 dwarf
• Expected ratio: 3:1 (i.e., 75 tall, 25 dwarf)

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.

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Worked Example 2 — Dihybrid Cross

A dihybrid cross (RrYy × RrYy) produces 640 offspring:

• Observed: 370 round yellow, 124 round green, 110 wrinkled yellow, 36 wrinkled green
• Expected ratio: 9:3:3:1

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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