AP Biology
Chi-Square Statistical Analysis
In science, rigorous tests are used to support the validity of results. The actual results of an experiment are unlikely to
match the expected results precisely. But how great a variance is significant? One way to decide is to use the Chisquare (χ2) test. This analytical tool tests the validity of a null hypothesis, which states that there is no statistically
significant difference between the observed results of an experiment and the expected results. When there is little
difference between the observed results and the expected results, you obtain a very low Chi-square value; thus your
hypothesis is supported.
You Must Know: What is meant by degrees of freedom; critical value, the null hypothesis; and how to perform Chisquare analysis of data.
Chi-square is calculated by subtracting the expected values from the observed values, squaring this number, and then
dividing by the expected value for all possible outcomes.
where:
 ∑ = sum of all possible outcomes
 o = observed number of individuals
 e = expected number of individuals
The Chi-square critical values table provides two values that you need to use to calculate Chi-square:
 Degrees of Freedom: this number is one less than the total number of possible outcomes for an experiment.
For example, in a monohybrid cross, there are two classes of offspring (dominant phenotype and recessive
phenotype). Therefore, there is just one degree of freedom. In a dihybrid cross, there are four possible classes
of offspring, so there are three degrees of freedom.
 Probability: the probability value (p) is the probability that a deviation as great as or greater than each Chisquare value would occur simply by chance. Many biologists agree that deviations having a chance probability
greater than 0.05 (5%) do not support the null hypothesis. Therefore, when you calculate Chi-square, you
should consult the table for the p value in the 0.05 row.
REMEMBER: If the calculated Chi-square value is greater than the critical value (from table), reject the null hypothesis.
If the Chi-square value is less than the critical value (from table), accept the null hypothesis. In other words, a large Chisquare value generally means that there is a statistical difference between your observed and expected values.
Sample Problem: Suppose you wish to determine experimentally whether or not the genes for eye color and wing shape
are located on the same chromosome or on different chromosomes in Drosophila melanogaster, the common fruit fly.
You have been given a vial containing a red-eyed male fruit fly with normal wings and a red-eyed female fruit fly with
normal wings. These are the F1 (first offspring) generation. After 2 weeks, you collect the offspring from this pair and
obtain the results shown in the figure below. On the basis of the results shown in Figure 2, what can you conclude about
the location of the genes with respect to each other?
Observed Phenotype and Numbers
F2 Results
Red Eyes and
Normal Wings
48
50
Males
Females
Red Eyes and
No Wings
13
9
Sepia Eyes and
Normal Wings
16
10
Sepia Eyes and
No Wings
4
10
Step 1: State a Null Hypothesis:
There is no statistically significant difference between the observed results and the expected results. In other words, the
genes for eye color and wing shape assort independently of one another, or are located on two different chromosomes.
Step 2: Determine the Expected Values:
Use a dihybrid Punnett square to determine your expected values from the cross described. In this scenario, it is likely
that the red-eyed/normal winged male and female in the F1 generation were heterozygous for both traits. Therefore,
the following cross was used: RrNn male x RrNn female (where R = red eyes; r = sepia eyes; N = normal wing; n = no
wing). Based on your null hypothesis, you are assuming that the 2 genes are located on separate chromosomes. The
Punnett square is shown below:
RN
Rn
rN
rn
RN
RRNN
RRNn
RrNN
RrNn
Rn
RRNn
RRnn
RrNn
Rrnn
rN
RrNN
RrNn
rrNN
rrNn
rn
RrNn
Rrnn
rrNn
rrnn
The Expected Results Are:
Red Eye, Normal Wing = 9/16 or 0.5625
Red Eye, No Wing = 3/16 or 0.1875
Sepia Eye, Normal Wing = 3/16 or 0.1875
Sepia Eye, No Wing = 1/16 or 0.0625
Step 3: Set Up a Data Chart:
The easiest way to keep track of your calculated numbers for the Chi-square is to use a data table like the one shown
below:
Phenotypes
Observed (o)
Red Eyes
Normal Wings
98
Red Eyes
22
No Wings
Sepia Eyes
26
Normal Wing
Sepia Eyes
14
No Wing
Total Flies Observed = 160
Expected (e)
160 x 0.5625 = 90
(o – e)
8
(o – e)2
(o – e)2 / e
64
0.711
160 x 0.1875 = 30
8
64
2.133
160 x 0.1875 = 30
4
16
0.533
160 x 0.0625 = 10
4
16
1.6
χ2 = 4.977
Step 4: Determine the Degrees of Freedom:
This is the number of categories (red eye/normal wing; red eye/no wing; sepia eye/normal wing; sepia eye/no wing)
minus one. For this data, the number of degrees of freedom is 3.
Step 5: Find the Probablility (p) Value for 3 Degrees of Freedom in the 0.05 Row:
This is the critical value. For this data, the critical value is 7.82.
Step 6: Accept or Reject the Null Hypothesis:
The null hypothesis stated that there was no statistically significant difference between the observed and expected data.
Since the χ2 value for this data is less than the critical value (4.977 < 7.82), you will accept the null hypothesis. This
means that you are accepting that the genes for eye color and wing shape assort independently of each other, or are
located on 2 different chromosomes…the genes are not linked.