Completion requirements
Standard Deviation
The standard deviation [1, 5, 12, 33, 35, 36, 37, 39] represents the average amount of
variability in a data set. In practical terms, the standard deviation is the average
distance from the mean. The larger the standard deviation, the larger the
average distance each value is from the mean of the data set.
The standard deviation is the most frequently used measure of variability.
The formula for calculating the standard deviation is:
\( s=\sqrt{\frac{\sum\left(x-\bar{x}\right)^2}{n-1}} \) (6.6)
\( x \) - each
individual value in the data set
\( \bar{x} \) - the
mean of all values in the data set
\( n \) - sample size
Procedure
- List each value. When calculating standard deviation, it is not necessary the values to be in any order.
- The mean of the data set is calculated.
- The mean is subtracted from each individual value.
- Each individual difference is squared.
- The sum all the squared deviations about the mean is calculated.
- The sum is divided by \( n-1 \) .
- Finally the square root is calculated.
Example
The data set [71, 50, 48, 67, 53] is given.
Each column of Table 6.3 represents a step from the procedure:
Table 6.3 Calculation of the standard deviation
|
\( x \) |
\( \bar{x} \) |
\( \left(x-\bar{x}\right) \) |
\( \left(x-\bar{x}\right)^2 \) |
|
71 |
57.8 |
13.2 |
174.24 |
|
50 |
57.8 |
-7.8 |
60.84 |
|
48 |
57.8 |
-9.8 |
96.04 |
|
67 |
57.8 |
9.2 |
84.64 |
|
53 |
57.8 |
-4.8 |
23.04 |
The column \( \left(x-\bar{x}\right) \) represents
the difference between the value and the mean, which is 57.8:
Fig. 6.12 Graphical representation of the data set and the deviations from mean
The sum of the deviations from the mean is always equal to zero.
\( \sum\left(x-\bar{x}\right)=0 \)
The squared deviations are used for escaping the negative sign, and when the sum is calculated it is no more 0:
\( \sum\left(x-\bar{x}\right)^2=438.8 \)
\( \frac{\sum\left(x-\bar{x}\right)^2}{n-1}=\frac{438.8}{4}=109.7 \)
And finally, the square root of the entire value is calculated, because it is necessary to return to the same units with which the calculation originally started.
\( s=\sqrt{\frac{\sum\left(x-\bar{x}\right)^2}{n-1}}=\sqrt{109.7}=10.47 \)