How would you explain the analysis of variance, assuming that your audience has not had a statistics class before? When one does a study of data, generally this implies an evaluation of the “mean” or average of that data. .i.e. What is the average time it takes a 5th grader to complete his final math exam?

A t test is used to test differences between two means. i.e. the mean of the experiment group vs a control group. An ANOVA test, on the other hand, is indicated when there are three or more means or populations to be examined.

When only two samples are looked at, the t test and ANOVA test will yield the same results.

Beyond two examples, the t test can be used to evaluate other means using many t tests, but this method becomes unreliable and subject to increased error.

ANOVA or analysis of variance allows one to use statistics to test the differences between two or more means and decreases the probability for a type 1 error, which might occur when looking at multiple two-sample t tests. This is why use of the ANOVA is indicated for testing hypotheses where there are multiple means or populations.

ANOVAs essentially takes the t test and applies it when testing or comparing three or more groups, means or variables.

Example experiment:

Study different types of protein consumption on weight gain. Is the effect of each type of protein consumption on weight gain significantly similar or different?

50 men total, 10 men per group, each group fed equal amount of protein calories but from different type of protein classification.

One way ANOVA – looking at effect of 5 different feeding strategies to increase weight gain. (i.e. high natural red meat protein diet, high vegetable protein diet, high protein powder diet, high fish protein diet, etc.). This is a one way ANOVA study because there is only one category (protein) that will be examined, as it relates to its effect on weight gain. It is balanced as there are an equal number of men (10) consuming the same type of protein. Simplistically put, the study will give us information as to whether or not each type of protein produces similar or different results (weight gain).

Two way ANOVA, as the name implies, includes the addition of a second variable that may be looked at as affecting weight gain. i.e. amount of exercise.

Three way ANOVA, as the name implies, includes the addition of a third variable. i.e. consumption of “other calories”.

In the end, the experiment may lead one to conclude that different types of protein consumption leads to weight gain. Unfortunately, experiments are never that cut and dry, and there are always other variables that must be accounted for. There is what’s called “error” due to “chance” and “beyond human control” or “variation” associated with “assignable causes”.

An F test must be used to compare these errors and/or variations, to determine if the population variances are the same or different. If the variances are the same or equal, the F ratio will be 1. This will allow one to complete their study / experiment and make final conclusions.

References:

https://explorable.com/anova

http://onlinestatbook.com/2/analysis_of_variance/intro.html

https://people.richland.edu/james/lecture/m170/ch13-f.html