Crosstab Software and Fischer’s Exact Test

Analyzing data in tables created by crosstab software can be done using a number of methods. One of those methods is Fischer’s Exact Test. This test lets you see if the proportions of one variable change when the proportions of another variable change. In other words, you’re testing to see if variables in your crosstab are truly independent.

Fischer’s Exact Test Explained

Let’s say you used crosstab software to create a contingency table that mapped out how many males and females preferred the color red over the color blue. You had 100 participants total: 50 males and 50 females.

Your crosstab may look something like this:

  Color Preference: Red vs. Blue
  Red Blue TOTAL
Male 37 13 50
Female 15 35 50
50 50 100

Fischer’s Exact Test could be used to determine if the preference for red over blue were somehow associated with the gender of the person. Here your null hypothesis would come into play, with the null hypothesis stating that the two numbers are not associated with or reliant upon each other in any way.

Fischer’s Exact Test would be used to compute the P value of the table, or the probability that the results are based on anything other than chance. The test calculates the exact probability of the table of cell data provided that:

  • The null hypothesis of independence is assumed to be true
  • The marginal totals of the observed tables are fixed

Both apply to the above crosstab outlining male and female color preferences, with fixed groups of 50 each and the underlying belief that the color preferences are not reliant upon gender.

Performing Fischer’s Exact Test

Advanced crosstab software and other programs can perform Fischer’s Exact Test for you based on the data you provide.

You can also perform the test manually, using the factorial mathematical operation to calculate the probability of the observed cell frequencies. Factorial is noted by the “!” symbol, which indicates multiplying a number by all integers smaller than the number. For example,

5! = 5*4*3*2*1 = 120

As noted, Fischer’s Exact Test only works when margins of a crosstab are fixed, as in the following table.

a b a + b
c d c + d
a + c b + d n


The equation to calculate Fischer’s Exact Test would then be:

(a + b)! * (c + d)! * (a + c)! * (b + d)!


n! * a! * b! * c! * d


The result would give you the P value of the observed cell frequencies in the table.

To calculate the P value of the male-female/red-blue crosstab, you can use the above equation or plug the data into a program that automatically computes Fischer’s Exact Test for you.

Fischer’s Exact Test in Action

Fischer’s Exact Test would give the P value of the red-blue crosstab, which is less than 0.0001. In this case, the P value indicates there is less than a .01 percent chance that the connection between color preference and gender is a result of chance.

This means the association between the rows and columns is indeed statistically significant. Once you know if the values in your crosstab are statistically significant or not, you can then move forward forming and testing further hypotheses and strategies as desired.