1. Chi-Square Analysis
Mendel’s Peas
and the
Goodness of Fit Test

2. We will develop the use of the χ2 distribution through an example from the history of biology.

3. In Austria in the mid 1800s, an Augustine monk, Gregor mendel, studied the garden pea and seven of its traits, such a shape and color of the peas, position of flowers on the plant, etc.
He is credited with discovering patterns of inheritance, the basis of the field of genetics.
Curiously, Mendel studied seven traits, one from each of the pea’s seven chromosomes. His theory of the independent assortment of genes occurs only when genes are on different chromosomes.

4. We will use one of Mendel’s studies, and some of his original data, to explore the χ2 test of significance.
Consider two different characteristics of peas, color and shape. The peas may be yellow or green, round or wrinkled.

5. If we cross a plant with yellow round peas with a plant having green wrinkled peas, and examine the progeny we will discover a uniform F1 generation.
The traits yellow and round are each dominant, while green and wrinkled are recessive. We use the letter Y for color, and R for pea shape, so the alleles are Y, y, R, and r.

6. This is a Punnett square to illustrate this dihybrid cross.
Yellow round pea
Green wrinkled pea
Notice the uniformity among the offspring, as all are YyRr.

7. YR Yr yR yr YYRR YYRr YyRR YyRr YYrr Yyrr yyRR yyRr yyrr gametes
Now we cross the F1 among themselves to produce the F2:
gametes YR Yr yR yr
YYRR
YYRr
YyRR
YyRr
YYrr
Yyrr
yyRR
yyRr
yyrr

8. YR Yr yR yr YYRR YYRr YyRR YyRr YYrr Yyrr yyRR yyRr yyrr gametes
Now we identify the yellow round peas:
gametes YR Yr yR yr
YYRR
YYRr
YyRR
YyRr
YYrr
Yyrr
yyRR
yyRr
yyrr

9. YR Yr yR yr YYRR YYRr YyRR YyRr YYrr Yyrr yyRR yyRr yyrr gametes
Now we identify the yellow wrinkled peas:
gametes YR Yr yR yr
YYRR
YYRr
YyRR
YyRr
YYrr
Yyrr
yyRR
yyRr
yyrr

10. YR Yr yR yr YYRR YYRr YyRR YyRr YYrr Yyrr yyRR yyRr yyrr gametes
Next we identify the green round peas:
gametes YR Yr yR yr
YYRR
YYRr
YyRR
YyRr
YYrr
Yyrr
yyRR
yyRr
yyrr

11. YR Yr yR yr YYRR YYRr YyRR YyRr YYrr Yyrr yyRR yyRr yyrr gametes
Finally, the last type of pea is green and wrinkled:
gametes YR Yr yR yr
YYRR
YYRr
YyRR
YyRr
YYrr
Yyrr
yyRR
yyRr
yyrr

12. Yellow round YYRR, YYRr, YyRR, YyRr Yellow wrinkled YYrr, Yyrr
So now we have four phenotypes (different physical forms) of peas originating from the single phenotype of the F1 generation.
They are, along with their genotypes and expected frequencies:
Yellow round
YYRR, YYRr, YyRR, YyRr
Yellow wrinkled
YYrr, Yyrr
Green round
yyRR, yyRr
Green wrinkled
yyrr

13. If Mendel’s understanding of genetics were correct, and the crosses made as he believed, the proportions of the four phenotypes should fit the calculations from the Punnet square.
Using the χ2 distribution, we are able to test to see if groups of individuals are present in the same proportions as expected.
This is rather like conducting multiple Z-tests for proportions at once.
In this example Mendel carried out the dihybrid cross to produce an F1 generation, and as expected, the F1 were all of the same phenotype, yellow and round.
Further, the F1 were crossed among themselves to produce the F2 generation. Mendel recorded the numbers of individuals in each category.

14. Observed Expected frequency Yellow round 315 Yellow wrinkled 101
The following table gives the observed numbers of each category.
Phenotype
Observed
Expected frequency
Yellow round
315
Yellow wrinkled
101
Green round
108
Green wrinkled 32

15. To make a χ2 test for “goodness of fit” we start as with all other tests of significance, with a null hypothesis.
Step 1:
H0: The F2 generation is comprised of four phenotypes in the proportions predicted by Mendelian genetics.
Ha: The F2 generation is not comprised of four phenotypes in the proportions predicted by Mendelian genetics.
Another way of saying this is that the null hypothesis claims the population fits our expected pattern, while the alternate hypothesis says it does not.

16. Assumptions: Our first assumption is that our data are counts
Assumptions: Our first assumption is that our data are counts. (We cannot use proportions or means.) With χ2, we do not always have a
sample of a population, and sometimes examine an entire population, as with this example. When working from a sample we must ensure that the sample is representative.
Step 2:
In order to check assumptions for this goodness of fit test we must calculate the expected counts for each category. Then we must meet two criteria:
1. All expected counts must be one or more.
2. No more than 20% of the counts may be less than 5.

17. We calculate the expected counts by finding the total number of observations and multiplying that by each expected frequency.
Phenotype
Observed
counts
Expected frequency
Expected counts
Yellow round
315
Yellow wrinkled
108
Green round
101
Green wrinkled 32

18. As you can see, all expected counts are greater than 5, so all assumptions are met.
Step 3:
The formula for the χ2 test statistic is:
where o = observed counts, and
e = expected counts
This calculation needs to be made in the graphing calculator.
Enter the observed counts in L1. Enter the expected frequencies in L2, as exact numbers. (Enter numbers like 1/3, directly, as fractions, never round to just .3 or .33.)

19. In L3 multiply L2 by 556. This will give the expected counts
In L3 multiply L2 by This will give the expected counts. The sum of L1 can be found using 1-Var Stats.
Now in L4, enter (L1-L3)2/L3, this will give you the χ2 contribution for each category.
Finally, χ2 is the sum of L4.
For this problem, the χ2 statistic is
In χ2, we always need to know and report the degrees of freedom. The degrees of freedom are the number of categories minus one.
Here we have 3 degrees of freedom.

20. Step 4:

21. Step 5:
The area can also be found with
cdf(.4700,10^99,3).
Fail to reject H0, as p = > α = .05.
Step 6:
We lack evidence that the pattern of pea phenotypes is different from expected. That is, the F2 generation are present in the expected proportions, 9:3:3:1.
Step 7:

22. Gregor Mendel did not have modern statistics to rely on for his data analysis, but none-the-less analyzed data in a way that led to this major scientific discovery, important to this day.
There has been speculation about his studies, or how he reported them, as the data is almost better than chance variation would produce.
He was, however, an Augustine monk, so perhaps he had a little help…