# Grand Mean: Definition

In ANOVA, the grand mean of a set of multiple subsamples is the mean of all observations: every data point, divided by the joint sample size.

If you don’t have information on all the data available to you, you can also:

1. Locate the subsample means in the ANOVA output (i.e. find the mean for each subsample),
2. Calculate a weighted mean, where each subsample mean is weighted by sample size.

If all the subsamples have the same number of points, the grand mean is simply a mean of the subsample means.

## Grand Mean Examples

Example 1: Find the grand mean for the following three subsamples:

• (6, 6, 3, 3)
• (1, 5, 0, 14)
• (9, 10, 11, 12)
• (0, 4, 0, 20).

Step 1: Find the sample means of each group:
5, 5, 8, and 6.
Step 2: Take the mean of the results from Step 1:
(5 + 5 + 8 + 6) / 4 = 24 / 4 = 6.

Example 2: For a slightly more complicated example, suppose your subsamples are of different sizes. You might have:

• (0, 4)
• (0, 3, 5, 0, 22, 0)
• (9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9).

The means are 2, 5, and 9, and to find the grand mean you cannot simply take the mean of those means. You need to either calculate weights or else take a mean over all data points.

If you choose the first method (calculate weights), multiply the mean by the number of data points, then divide by the total number of points:
(2 * 2 + 5 * 6 + 9 * 11)/ 14 = 9.5.
See finding a weighted mean for more info on weights, and why you would want to calculate them.

The second method will give you the same answer, but you would treat all the data as one big sample: (0, 4, 0, 3, 5, 0, 22, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9). Adding all those numbers up and dividing by 19 gives you 9.5.

If you are rusty on finding sample means, you may find this video helpful:

## References

• Hanlon & Larget. Analysis of Variance.