THE ESSENTIAL STATS 25
Probability that chance alone gave us a chi-squared greater than or equal to 0.97
is 0.6241
Based on these results we conclude that wealth has no signiﬁcant affect on health (remember
that our observed values are made up!).
4.4 Fisher’s Exact Test
4.4.1 Why and when
Fisher’s exact test can be used instead of chi-squared when you have two variables (for example
health and wealth), each having two categories (for example: sick, healthy and poor, rich), and
one or more of the expected counts for the four possible categories (2 · 2 = 4) are below 10
(remember: you cannot use chi-squared if even one expected count is less than 10). Like chi-
squared, Fisher’s exact test can be used to see if there is a relationship between the two variables
as well as to measure the deviation of observed data from expectation.
Reputedly, Fisher was inspired to develop this test after an acquaintance, Dr. Muriel Bristol,
claimed to be able to distinguish between a cup of tea in which the milk was poured before the tea
and a cup in which the tea was poured before the milk
1
. In honor of Dr. Bristol, we will test this
hypothesis.
4.4.2 Calculate with Example
Say we have collected the following data:
The above generalizes to:
If our tea taster really can correctly identify which cups of tea had the milk poured ﬁrst and
which did not, then we expect a and d to be large (meaning he or she guessed correctly most of the
time) and b and c to be small (incorrect guesses). If b and c are large, then our taster consistently
misidentiﬁed the tea. If our taster consistently misidentiﬁes the tea then he or she probably can dis-
tinguish between the two types, but has labeled them incorrectly. If our taster cannot distinguish
1. http://en.wikipedia.org/wiki/Muriel_Bristol
Actual Order
Milk First Tea First Total
Claim of Tea
Taster
Milk First 3 1 4
Tea First 2 4 6
Total 5 5 10
Variable A
Category
A1
Category
A2
Total
Variable B
Category B1 a b a + b
Category B2 c d c + d
Total a + c b + d n

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