Statistics - Grand Mean

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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.

Jackson167104
Thomas528146
Garrard829127

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 }$
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