Chi squared in genetics

Laura Armstrong & Joe Wolfensohn

Teachers

Laura Armstrong Joe Wolfensohn

Recall Questions

This topic requires prior knowledge of expected phenotypic ratios from monohybrid and dihybrid crosses and a foundational understanding of probability in genetic outcomes. You can test your knowledge on these below.

What is a phenotypic ratio?

What is the expected ratio in a heterozygous dihybrid genetic cross?

What does it mean if observed and expected values are different?

Topic Explainer Video 1

Check out this @JoeDoesBiology video that explains mutations, or read the full notes below. Once you've gone through the whole note, try out the practice questions!

Topic Explainer Video 2

Check out this @JoeDoesBiology video that explains mutations, or read the full notes below. Once you've gone through the whole note, try out the practice questions!

What is the Chi-Squared Test?

  • A statistical test used to compare observed results from a genetic cross with expected ratios.

  • It helps determine if any difference is due to chance or if the difference is significant.

When Do We Use It?

  • When investigating inheritance patterns in monohybrid or dihybrid crosses.

  • To test whether observed results fit a Mendelian ratio, such as 3:1 or 9:3:3:1.

  • It is used when you have categoric data - for example different phenotypes.

The Chi-Squared Formula

x2=(OE)2Ex^{2}=\sum\frac{(O-E)^{2}}{E}

  • O = observed frequency

  • E = expected frequency

Steps to Carry Out the Chi-Squared Test

  1. State your null hypothesis (H₀)
    → e.g., “There is no significant difference between observed and expected results.”
  2. Calculate expected values
    → Based on the predicted ratio and total number of offspring.

  3. Apply the Chi-squared formula

  4. Calculate degrees of freedom (df):
    → df= n − 1
    → n = number of categories (phenotypes)

  1. Compare χ² value to the critical value at p = 0.05

  2. Draw a conclusion:

    • If χ² is less than the critical value → accept H₀ (difference between observed and expected is not significant and there is more than 5% probability the difference is due to chance)

    • If χ² is greater than or equal to the critical value → reject H₀ (difference between observed and expected is significant and less than 5% probability the difference is due to chance)

Worked Example

Question: In a dihybrid cross, you expect a 9:3:3:1 ratio. Out of 160 pea plants, you observe:

  • Yellow round = 63

  • Yellow wrinkled = 29

  • Green round = 36

  • Green wrinkled = 32

 

Step 1: Expected values (based on 9:3:3:1 ratio)

  • 9/16 × 160 = 90

  • 3/16 × 160 = 30

  • 3/16 × 160 = 30

  • 1/16 × 160 = 10

Step 2: χ² calculation

 

Phenotype

O

E

O - E

(O-E)2

(O-E)2/E

Yellow round

63

90

-27

729

8.1

Yellow wrinkled

29

30

-1

1

0.03

Green round

36

30

6

36

1.2

Green wrinkled

32

10

22

482

48.2

         

Σ = 57.5

 

Step 3: 

Degrees of freedom = 4 - 1 = 3
 

Critical Values table

 

 

P Value

df

0.1

0.05

0.01

1

2.706

3.841

6.635

2

4.605

5.991

9.21

3

6.251

7.815

11.35

4

7.779

9.488

13.28

5

9.236

11.07

15.09

6

10.65

12.59

16.81

 

Critical value at p = 0.05 = 7.815

Conclusion:
Since 57.5 > 7.815, we reject the null hypothesis.

The difference between observed and expected is significant and there is a less than 5% probability the difference is due to chance.

This suggests that the genes may be linked.

Key Terms

  • Null hypothesis (H₀): Assumes no significant difference between observed and expected results
  • Critical value: A threshold value from a statistical table.
  • Degrees of freedom: One less than the number of phenotypic categories for Chi squared.

Exam Tips

  • Always show working for full marks, including expected values and degrees of freedom.

  • Don't say the results are significant. Always say the difference is or isn't significant.

  • Use the specific structure in conclusions to ensure you get the marks.

  • Link findings back to biology — e.g., genes may be linked on the same chromosome.

A student crosses two heterozygous fruit flies for two genes and expects a 9:3:3:1 ratio. The observed numbers of offspring are:

  • Red eyes, normal wings = 86

  • Red eyes, vestigial wings = 32

  • White eyes, normal wings = 27

  • White eyes, vestigial wings = 15

Test whether the results are significantly different from the expected ratio using a Chi-squared test. Use a significance level of 0.05. (5 marks)

Practice Question

Try to answer the practice question from the TikTok on your own, then watch the video to see how well you did!