Chi-Squared Guide

The chi-squared test allows for scientists to determine how significant results are from what they expect. In experiments, it is not always easy to see whether your results are due to random chance, or due to certain biological factors that you are testing, so they created the chi-squared test as a uniform measuring tool to go by. This test will determine whether you have a significant difference between your observed and expected results.

Method for Genetics:

1. Gather data on your cross. This should be the number of offspring you have in a specific cross for each phenotype.

2. Calculate the observed frequency, which is the percentage of the population that each phenotype made.

3. Calculate the expected frequency, which is the percentage of the population that you would expect each phenotype to have.

The default in an expected frequency would be that all phenotypes would be equal, unless you run the Punnett Square and it tells you otherwise.

4. Determine the number of degrees of freedom (df), which is one less than the total number of phenotypes. (e.g. if there are 4 phenotypes in your offspring, then df = 4-1 = 3).

5. Using the df that you have calculated, find the critical region for chi-squared from a table of chi-squared values for the significance level (p) of 0.05. The critical region is any value greater than the value in the table.

6. Calculate the chi-squared using this equation:

O = ObservedΣ= the sum of

E = Expected

7. Compare the value of the chi-squared you calculated with the critical region of the chart.

You can then make one of two conclusions:

-If your calculated value is in the critical region, the difference between expected and observed are significant. This means that the two genes are linked

-If the calculated value is not in the critical region, the difference is not statistically significant. This means the genes are unlinked and the difference is due to random chance.

Example Problem:

White leghorn chickens with large single combs were crossed with indian game fowl with dark feathers and small pea combs. All of the F1 crosses were white with pea combs. They were crossed together, so the Expected ratio was 9:3:3:1. The observed results are shown below:

White/Pea / White/Single / Dark/Pea / Dark/Single / Total
Observed / 111 / 37 / 34 / 8 / 190
Expected / 9/16 x 190 = 106.9 / 3/16 x 190 = 35.6 / 3/16 x 190 = 35.6 / 1/16 x 190 = 11.9 / 190

Degrees of freedom = 4-1 = 3

Critical value at 3 degrees of freedom is 7.815.

Chi-Squared = (111-106.9)2/106.9 + (37 – 35.6)2/35.6 + (34 – 35.6)2/35.6 + (8 – 11.9)2/11.9

= 1.56

Conclusion: Since 1.56 is less than the critical value of 7.815, the difference between the expected and observed results is not statistically significant.

In other words: The results are close enough to the expected 9:3:3:1 ratio that the traits are unlinked and assort independently.