Fisher's Exact Probability Test

Fisher's Exact Probability Test

This page will perform the Fisher exact probability test for a table of frequency data cross-classified according to two categorical variables, A and B, each of which has two levels or subcategories. To proceed, enter the values of A₁B₁, A₁B₂, etc., into the designated cells. (If you start with cell A₁B₁, pressing the "Tab" key will move you through the other data-entry cells.) Do not enter anything into the cells labeled "Totals"; these values will be calculated automatically. When all cell values have been entered, click the "Calculate" button, then scroll down the page to view the results of the calculation. To perform a new analysis with a new set of data, click the "Reset" button.

A brief description of the logic and computational procedure underlying the Fisher test will be found toward the bottom of this page. A fuller account is found in Subchapter 8a of Concepts and Applications.

Although the Fisher test is designed for use with relatively small samples, the programming for this page will actually handle fairly large samples, up to about n = 100, depending on how the frequencies are arrayed within the four cells.

Data Entry:

	B1	B2	Totals
A1
A2
Totals

Probability:

one-tail:
two-tail:

Logic and Procedure:
Consider a 2x2 contingency table of the sort described above, with the cell frequencies represented by a, b, c, d, and the marginal totals represented by a+b, c+d, a+c, b+d, and n.

	B1	B2	Totals
A1	a	b	a+b
A2	c	d	c+d
Totals	a+c	b+d	n

If there were no systematic association between the variables A and B within the population from which the cell frequencies are randomly drawn, the probability of any particular possible array of cell frequencies, a, b, c, d, given fixed values for the marginal totals a+b, c+d, etc., would be given by the hypergeometric rule

which for computational purposes reduces to

Also, the degree of disproportion within any array of cell frequencies—in effect, the degree of ostensible association between variables A and B within the sample—can be measured by the absolute difference

For any particular observed array of cell frequencies, the programming embedded in this page calculates the probability of that particular array plus the probabilities of all other possible arrays whose degree of disproportion is equal to or greater than that of the observed array. Thus, for the observed array

2	7	9
8	2	10
10	9	19

the one-tailed probability would be the sum of the separate probabilities for the arrays

		probability
2	7
8	2	0.01754
1	8
9	1	0.00097
0	9
10	0	0.00001

sum = 0.01852			(one-tailed probability)

And the two-tailed probability would be that sum plus the sum of the separate probabilities for the arrays of equal or greater disproportion at the other extreme:

		probability
8	1
2	8	0.00438
9	0
1	9	0.00011

sum = 0.00449

two-tailed probability = 0.01852 + 0.00449 = 0.02301

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