Fisher's exact test is a significance test for a contingency table that checks whether two categorical variables are associated, especially when sample sizes are small. In Intro to Statistics, it gives an exact p-value instead of relying on a chi-square approximation.
Fisher's exact test is the small-sample test you use in Intro to Statistics when you have two categorical variables in a contingency table and the chi-square test is shaky. It asks whether the counts you observed would be unlikely if the null hypothesis of independence were true.
The big idea is that the test looks at the table as a whole, not at one cell in isolation. In a 2 by 2 table, it uses the row totals and column totals as fixed and works out the probability of getting the observed arrangement, or one even more extreme, under the assumption of no association. That is why it is called "exact": the p-value comes from the actual distribution of possible tables, not from a large-sample shortcut.
This is where the hypergeometric distribution shows up. Fisher's exact test is built on the same without-replacement logic, so it fits situations where counts come from a small finite set and each count affects the others. If you sample a handful of people, organisms, or items, the test can still give a valid result even when expected cell frequencies are below 5.
A common intro-stats example is a tiny clinical or classroom study, like comparing a new study method versus the usual method with pass or fail outcomes. If one or more cell frequencies are small, the chi-square approximation can be unreliable, so Fisher's exact test is the cleaner choice. You still compare the p-value to your significance level, but the p-value is coming from exact table probabilities rather than a normal approximation.
One thing to watch for is that Fisher's exact test does not tell you how strong the relationship is. It only tells you whether the table is surprising enough under the null hypothesis. If you want strength of association, you would look at a measure like Phi Coefficient or Cramer's V after, or alongside, the test.
Fisher's exact test matters because Intro to Statistics is full of questions where the data set is too small for the usual chi-square routine. If a problem gives you a 2 by 2 contingency table with tiny counts, you need to know whether to trust a chi-square p-value or switch to Fisher's exact test.
It also connects several course ideas that show up together: contingency tables, the null hypothesis, observed frequency, and p-value. Instead of treating a table like a bunch of isolated counts, Fisher's exact test asks you to think about the whole pattern of counts under independence.
That skill shows up in homework and quizzes where you interpret whether two categorical variables seem related. It also shows up in written responses when you explain why a test is appropriate, especially if the sample is small or a cell frequency is below 5. Knowing Fisher's exact test keeps you from using a method that is only approximate when the data need something exact.
It is also a good bridge to the hypergeometric distribution, which is one of the few probability models in intro stats that feels directly tied to a test procedure. Once you see that connection, the test stops looking like a random formula and starts looking like a logical extension of sampling without replacement.
Keep studying Intro to Statistics Unit 11
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view galleryContingency Table
Fisher's exact test is run on a contingency table, usually a 2 by 2 table in Intro to Statistics. The table gives the observed frequency counts for two categorical variables, and the test checks whether that arrangement looks unusual under independence. If you cannot read the table correctly, you cannot set up the test correctly.
Null Hypothesis
The null hypothesis for Fisher's exact test says the two categorical variables are independent, meaning no association in the population. The p-value tells you how surprising your observed table would be if that null were true. This makes the test a direct hypothesis test, not just a count comparison.
Hypergeometric Distribution
Fisher's exact test is built on hypergeometric probability, which is why it works well for small samples and without-replacement situations. The distribution describes the chance of getting particular cell counts when the margins are fixed. That same logic is what gives Fisher's test its exact p-value.
P-value
The output you usually interpret from Fisher's exact test is the p-value. A small p-value means the observed table, or one more extreme, would be unlikely if the null hypothesis of independence were true. In class problems, this is the number you compare to your chosen significance level.
A quiz or problem set question usually gives you a small 2 by 2 contingency table and asks whether the data show an association between two categorical variables. Your job is to decide if Fisher's exact test is the right procedure, especially when one or more expected cell frequencies are small. Then you interpret the p-value in context, using the null hypothesis of no association.
You may also be asked to explain why chi-square is not the best choice here. The usual move is to say the sample is small, so the chi-square approximation may not be reliable. If the problem gives you software output, you read the Fisher p-value and decide whether it is small enough to reject the null hypothesis.
These two tests both look at whether two categorical variables are associated in a contingency table, but they are not used the same way. Chi-square is an approximation that works best with larger samples and sufficiently large expected cell frequencies. Fisher's exact test is the safer choice for small samples, especially in a 2 by 2 table.
Fisher's exact test checks whether two categorical variables in a contingency table are associated when the sample is too small for a reliable chi-square approximation.
The test gives an exact p-value based on the possible tables under the null hypothesis of independence.
It is especially useful for 2 by 2 tables with small expected cell frequencies or very small sample sizes.
A small p-value means the observed table would be unlikely if the variables were truly independent.
Fisher's exact test tells you about statistical significance, not the size or strength of the relationship.
Fisher's exact test is a hypothesis test for a contingency table that checks whether two categorical variables are independent. In Intro to Statistics, you use it when the sample is small and the chi-square test may not be accurate. It produces an exact p-value from the table counts.
Use Fisher's exact test when your contingency table has small counts, especially in a 2 by 2 table. Chi-square relies on a large-sample approximation, so it can be shaky when expected cell frequencies are low. Fisher's test handles that small-sample situation directly.
No, it mainly tests whether the relationship is statistically significant under the null hypothesis of independence. A small p-value says the table is unlikely if there were no association. If you want strength, you look at a measure like Phi Coefficient or Cramer's V.
It is called exact because the p-value comes from the actual probability of the observed table, not from a normal or chi-square approximation. That makes it reliable for small samples. The calculation is based on the exact distribution of tables with the same margins.