# Statistics - Grand Mean

When sample sizes are equal, in other words, there could be five values in each sample, or n values in each sample. The grand mean is the same as the mean of sample means.

## Formula

${X_{GM} = \frac{\sum x}{N}}$

Where −

• ${N}$ = Total number of sets.

• ${\sum x}$ = sum of the mean of all sets.

### Example

Problem Statement:

Determine the mean of each group or set's samples. Use the following data as a sample to determine the mean and grand mean.

 Jackson Thomas Garrard 1 6 7 10 4 5 2 8 14 6 8 2 9 12 7

Solution:

Step 1: Compute all means

${M_1 = \frac{1+6+7+10+4}{5} = \frac{28}{5} = 5.6 \\[7pt] \, M_2 = \frac{5+2+8+14+6}{5} = \frac{35}{5} = 7 \\[7pt] \, M_3 = \frac{8+2+9+12+7}{5} = \frac{38}{5} = 7.6 }$

Step 2: Divide the total by the number of groups to determine the grand mean. In the sample, there are three groups.

${X_{GM} = \frac{5.6+7+7.6}{3} = \frac{20.2}{3} \\[7pt] \, = 6.73 }$