Arithmetic Mean
The mean is the average value of the distribution.
The arithmetic mean is the value obtained by adding all the data and dividing the result by the total number of data.
is the symbol of the arithmetic mean.
Example
The weight of six children can be expressed by the following values: 84, 91, 72, 68, 87 and 78 pounds. Find the mean.
Arithmetic Mean for Grouped Data
If the data is grouped in a frequency table, the expression of the mean is:
Example
The test scores of 42 students are shown in the table below. Calculate the mean.
x_{i} | f_{i} | x_{i} · f_{i} | |
---|---|---|---|
[10, 20) | 15 | 1 | 15 |
[20, 30) | 25 | 8 | 200 |
[30,40) | 35 | 10 | 350 |
[40, 50) | 45 | 9 | 405 |
[50, 60 | 55 | 8 | 440 |
[60,70) | 65 | 4 | 260 |
[70, 80) | 75 | 2 | 150 |
42 | 1 820 |
Properties of the Arithmetic Mean
1. The sum of the deviations of all values of a distribution from their arithmetic mean is zero.
The sum of the deviations of the numbers 8, 3, 5, 12, 10 of the arithmetic mean 7.6 is equal to 0:
8 − 7.6 + 3 − 7.6 + 5 − 7.6 + 12 − 7.6 + 10 − 7.6 =
= 0. 4 − 4.6 − 2.6 + 4. 4 + 2. 4 = 0
2. The sum of the squares of the deviations of the values of the variable with respect to any number is minimized when the number matches the arithmetic mean.
3. If all values of the variable are added by the same number, the arithmetic mean is increased by that number.
4. If all values of the variable are multiplied by the same number, the arithmetic average is multiplied by that number.
Observations on the Arithmetic Mean
1. The average can be found only in quantitative variables.
2. The mean is independent of the widths of the classes.
3. The mean is very sensitive to extreme scores. For example, if there is a distribution with the following values:
65, 69, 65, 72, 66, 75, 70, 110.
The mean is equal to 74, which is a measure of centralization unrepresentative of the distribution.
4 The mean cannot be calculated if there is a class with an indeterminate width.
x_{i} | f_{i} | |
---|---|---|
[60, 63) | 61.5 | 5 |
[63, 66) | 64.5 | 18 |
[66, 69) | 67.5 | 42 |
[69, 72) | 70.5 | 27 |
[72, ∞ ) | 8 | |
100 |
In this case it is not possible to find the average because the last class mark cannot be calculated.