Chi-Square Test

Chi-Square Test Explained

Example Scenario: Plant Phenotypes

• A plant with tall and short phenotypes is used to demonstrate the Chi-Square test.
• The expected Mendelian ratio is 3:1 (75% tall, 25% short).

Observed vs. Expected Values

• Observed Values: Actual counts from the garden.
  • Example: 58 tall plants, 22 short plants (total 80 plants).
• Expected Values: Calculated based on the expected ratio.
  • If 80 plants are observed with a 3:1 ratio:
    • Tall: $0.75 \times 80 = 60$
    • Short: $0.25 \times 80 = 20$

Chi-Square Formula

• The Chi-Square formula is: $\chi^2 = \sum{\frac{(O - E)^2}{E}}$, where:
  • $O$ = Observed value
  • $E$ = Expected value
  • $\sum$ = Summation across all categories

Calculation

• For Tall Plants:
  • $(O - E)^2 = (58 - 60)^2 = (-2)^2 = 4$
  • $\frac{(O - E)^2}{E} = \frac{4}{60} = 0.0667$
• For Short Plants:
  • $(O - E)^2 = (22 - 20)^2 = (2)^2 = 4$
  • $\frac{(O - E)^2}{E} = \frac{4}{20} = 0.2$
• Chi-Square Value:
  • $\chi^2 = 0.0667 + 0.2 = 0.2667$

Degrees of Freedom

• Degrees of freedom (df) = Number of categories - 1
• In this example: df = 2 (tall, short) - 1 = 1

Using the Chi-Square Table

• A chi-squared table is used to determine the p-value.
• A significance level (alpha) of 0.05 is commonly used.
• With df = 1 and alpha = 0.05, the critical value from the table is 3.841.

Interpretation

• Compare the calculated Chi-Square value (0.2667) to the critical value (3.841).
• If the Chi-Square value is less than the critical value, accept the hypothesis.
• If the Chi-Square value is greater than the critical value, reject the hypothesis.
• In this case: 0.2667 < 3.841, so we accept the hypothesis.

Conclusion

• The difference between observed and expected values is small enough to be explained by random chance.
• There is no significant evidence to reject the expected 3:1 Mendelian ratio.