Statistics » Mean deviation from the mean

The mean deviation from the mean is a measure of spread. It tells you how far the values deviate from the mean. Remember that the deviation always is a positive number.

You can calculate the mean deviation from the mean as follows:
- Calculate the mean.
- Calculate the deviation for every value.
- Calculate the mean of those deviations.

Example 1

Given is the row: 2,  3,  3,  4,  5,  6,  6,  6,  6,  7
Calculate the mean deviation from the mean.

Answer
Mean = 2 + 3 + 3 + 4 + 5 + 6 + 6 + 6 + 6 + 710 = 4.8
Deviations are:
4.8 – 2 = 2.8
4.8 – 3 = 1.8
4.8 – 3 = 1.8
4.8 – 4 = 0.8
5 – 4.8 = 0.2
6 – 4.8 = 1.2
6 – 4.8 = 1.2
6 – 4.8 = 1.2
6 – 4.8 = 1.2
7 – 4.8 = 2.2
Mean deviation from the mean =
2.8 + 1.8 + 1.8 + 0.8 + 0.2 + 1.2 + 1.2 + 1.2 + 1.2 + 2.210 = 1.36

Example 2

Given are the following numbers: 459, 65, 5, 149, 759, 109, 15.
Calculate the mean deviation from the mean.

Answer
Mean = 459 + 65 + 5 + 149 + 759 + 109 + 157 = 223
Deviations are:
459 – 223 = 236
223 – 65 = 394
223 – 5 = 158
223 – 149 = 74
759 – 223 = 536
223 – 109 = 114
223 – 15 = 208
Mean deviation from the mean =
236 + 158 + 218 + 74 + 536 + 114 + 2087 = 22047 ≈ 220.6

Example 3

A teacher has made a frequency table of marks given for a written test. Calculate the mean deviation from the mean. Round to three decimals.

mark 4 5 6 7 8 9 10
frequency 2 4 4 9 8 2 1

Answer
There are 2 + 4 + 4 + 9 + 8 + 2 + 1 = 30 pupils.
Mean = 4 × 2 + 5 × 4 + 6 × 4 + 7 × 9 + 8 × 8 + 9 × 2 + 1030 = 6.9
Deviations are:
6.9 – 4 = 2.9
6.9 – 5 = 1.9
6.9 – 6 = 0.9
7 – 6.9 = 0.1
8 – 6.9 = 1.1
9 – 6.9 = 2.1
10 – 6.9 = 3.1
Remember that these deviations are happening with the same frequency as in the frequency table. The frequency table, with the deviations, will look like this:

mark 4 5 6 7 8 9 10
deviation 2.9 1.9 0.9 0.1 1.1 2.1 3.1
frequency 2 4 4 9 8 2 1

Mean deviation from the mean =
2.9 × 2 + 1.9 × 4 + 0.9 × 4 + 0.1 × 9 + 1.1 × 8 + 2.1 × 2 + 3.130 ≈ 1.729


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