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Chi squared in genetics

Laura Armstrong & Joe Wolfensohn

Teachers

Laura Armstrong Joe Wolfensohn

Contents

What is the Chi-Squared Test?

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Steps to Carry Out the Chi-Squared Test

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Worked Example

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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?

The ratio of observed physical traits (phenotypes) in offspring from a genetic cross.

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

9:3:3:1

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

It may suggest that the genes are not assorting independently - possibly due to linkage. It could also be the result of epistasis.

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

  • 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.
No answer provided.

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.

No answer provided.

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)

  • Expected values (for 9:3:3:1 ratio from 160 total):

    • 9/16 × 160 = 90

    • 3/16 × 160 = 30

    • 3/16 × 160 = 30

    • 1/16 × 160 = 10

  • χ² calculation:

Phenotype

O

E

O - E

(O-E)2

(O-E)2/E

Red eyes, normal wings

86

90

-4

16

0.17

Red eyes, vestigial wings

32

30

2

4

0.13

White eyes, normal wings

37

30

7

49

1.63

White eyes, vestigial wings

15

10

5

25

2.5
         

Σ = 3.11

 

  • Degrees of freedom = 4 - 1 = 3

  • Critical value at p = 0.05 = 7.815

Conclusion:

  • χ² value (3.11) is less than 7.815

  • Accept the null hypothesis

  • No significant difference between observed and expected, more than 5% probability that the difference is due to chance.

Practice Question

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

Epistasis