Many times when we study a group, we are really comparing two populations. Depending upon the parameter of this group we are interested in and the conditions we are dealing with, there are several techniques available. Statistical inference procedures that concern the comparison of two populations cannot usually be applied to three or more populations. To study more than two populations at once, we need different types of statistical tools. Analysis of variance, or ANOVA, is a technique from statistical interference that allows us to deal with several populations.

## Comparison of Means

To see what problems arise and why we need ANOVA, we will consider an example. Suppose we are trying to determine if the mean weights of green, red, blue and orange M&M candies are different from each other. We will state the mean weights for each of these populations, μ_{1}, μ_{2}, μ_{3} μ_{4} and respectively. We may use the appropriate hypothesis test several times, and test C(4,2), or six different null hypotheses:

- H
_{0}: μ_{1}= μ_{2}to check if the mean weight of the population of the red candies is different than the mean weight of the population of the blue candies. - H
_{0}: μ_{2}= μ_{3}to check if the mean weight of the population of the blue candies is different than the mean weight of the population of the green candies. - H
_{0}: μ_{3}= μ_{4}to check if the mean weight of the population of the green candies is different than the mean weight of the population of the orange candies. - H
_{0}: μ_{4}= μ_{1}to check if the mean weight of the population of the orange candies is different than the mean weight of the population of the red candies. - H
_{0}: μ_{1}= μ_{3}to check if the mean weight of the population of the red candies is different than the mean weight of the population of the green candies. - H
_{0}: μ_{2}= μ_{4}to check if the mean weight of the population of the blue candies is different than the mean weight of the population of the orange candies.

There are many problems with this kind of analysis. We will have six *p*-values. Even though we may test each at a 95% level of confidence, our confidence in the overall process is less than this because probabilities multiply: .95 x .95 x .95 x .95 x .95 x .95 is approximately .74, or an 74% level of confidence. Thus the probability of a type I error has increased.

At a more fundamental level, we cannot compare these four parameters as a whole by comparing them two at a time. The means of the red and blue M&Ms may be significant, with the mean weight of red being relatively larger than the mean weight of the blue. However, when we consider the mean weights of all four kinds of candy, there may not be a significant difference.

## Analysis of Variance

To deal with situations in which we need to make multiple comparisons we use ANOVA. This test allows us to consider the parameters of several populations at once, without getting into some of the problems that confront us by conducting hypothesis tests on two parameters at a time.

To conduct ANOVA with the M&M example above, we would test the null hypothesis H_{0}:μ_{1} = μ_{2} = μ_{3}= μ_{4}. This states that there is no difference between the mean weights of the red, blue and green M&Ms. The alternative hypothesis is that there is some difference between the mean weights of the red, blue, green and orange M&Ms. This hypothesis is really a combination of several statements H_{a}:

- The mean weight of the population of red candies is not equal to the mean weight of the population of blue candies, OR
- The mean weight of the population of blue candies is not equal to the mean weight of the population of green candies, OR
- The mean weight of the population of green candies is not equal to the mean weight of the population of orange candies, OR
- The mean weight of the population of green candies is not equal to the mean weight of the population of red candies, OR
- The mean weight of the population of blue candies is not equal to the mean weight of the population of orange candies, OR
- The mean weight of the population of blue candies is not equal to the mean weight of the population of red candies.

In this particular instance, in order to obtain our p-value, we would utilize a probability distribution known as the F-distribution. Calculations involving the ANOVA F test can be done by hand, but are typically computed with statistical software.

## Multiple Comparisons

What separates ANOVA from other statistical techniques is that it is used to make multiple comparisons. This is common throughout statistics, as there are many times where we want to compare more than just two groups. Typically an overall test suggests that there is some sort of difference between the parameters we are studying. We then follow this test with some other analysis to decide which parameter differs.