📂Statistical Test

What is Analysis of Variance or ANOVA in Statistics?

Definition ^{1 2}

Analyzing variance to compare the population means of two or more groups is called analysis of variance, and it’s often abbreviated as ANOVA.

Explanation

Intuitively, one might think that comparing the sample means would suffice to compare population means, but in the realm of statistics, simply comparing numerical values has little significance.

Mean of A	Mean of B
0.0142	0.0271

For example, consider the histograms of samples obtained from two groups, $A$ and $B$, as above. Mathematically, the sample mean of group $A$ is $.0142$, and that of group $B$ is $.0271$, with group $B$ having a marginally larger value. However, merely comparing the values in this way is something even a child could do, and if asked whether there is a true difference in the population means of the two groups, it is not easy to readily say yes.

Now, consider data with almost the same sample means but with lower variance, more tightly clustered around their respective population means. Although the difference in sample means is almost the same as the previous scenario, the change in variance allows one to confidently discuss the differences between the population means.

This captures the basic concept of ‘analyzing variance to compare means’ in variance analysis. Actual variance analysis is mainly divided into one-way ANOVA and two-way ANOVA, and when there are $k$ populations, the null hypothesis is set as follows to determine if all $k$ population means are equal:

• $H_{0}$: $\mu_{1} = \cdots = \mu_{k}$
• $H_{1}$: At least one $\mu_{i}$ differs from the other population means.