7.1.6 The Chi-squared Test

Chi-squared test

The chi-squared test is a statistical method used to determine whether the difference between observed results and expected results is small enough to be due to chance. It helps evaluate whether observed data supports the null hypothesis.

Null Hypothesis

The null hypothesis states that there is no statistically significant difference between observed and expected results. For example, "there is no difference between the observed number of heads and tails when flipping a coin."

Criteria for Using the Chi-Squared Test

• The sample size must be large enough (at least 20).
• Data must fall into discrete categories (e.g., phenotypes such as red or white flowers).
• Only raw counts can be used (not percentages, ratios, or rates).

Formula

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

Where:

• $OO$ = Observed value
• $EE$= Expected value

Steps for Performing the Chi-Squared Test

1. Calculate the expected values based on the genetic ratio (e.g., a 3:1 ratio).

2. Subtract the observed value from the expected value and square the result: . $(O - E)^2$

3. Divide the squared difference by the expected value for each category.

4. Sum all the values to get the chi-squared value $(\chi^2)$.

Comparing the Chi-Squared Value to the Critical Value

1. Calculate the degrees of freedom:

$$\text{Degrees of freedom} = \text{Number of categories} - 1$$

2. Use the p-value of 0.05 to find the critical value from a chi-squared table.
3. Compare the calculated chi-squared value to the critical value:

• If $χ2≥critical value\chi^2 \geq \text{critical value}$, accept the null hypothesis (difference is due to chance).
• If $χ2<critical value\chi^2 < \text{critical value}$, reject the null hypothesis (difference is significant).

Example

• Observed: 90 round seeds, 30 wrinkled seeds.
• Expected ratio: 3:1 (round: wrinkled).
• Total = 120 seeds, Expected: 90 round (75%) and 30 wrinkled (25%).
• Use the formula to calculate $\chi^2$, compare it to the critical value, and evaluate the null hypothesis.