The standard deviation reflects the average of the differences between the individual results and their average, but it is not identical to this average, as we shall show. In order to avoid confusion, we denote the average of the results by the letter M. The greater the gap between the individual results and the M, the greater the standard deviation.
In other words, the more widely the results are dispersed around the average (M), the greater the standard deviation. The difference between the results and the average is always measured as a positive value. It makes no difference whether a datum is to the left or to the right of the average.
The standard deviation is not the average of the differences, although in many cases it is very close to the average of the differences, or is perhaps even equal to it. The standard deviation is calculated as follows (as in the previous example):
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We take the square of the differences between the individual results and the average.
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We calculate the average of the squares.
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We calculate the square root of the averages of the squares (taking the square root is designed to eliminate the effect of squaring the differences).
This series of operations results in differences between the standard deviation and the average of the differences.
We will highlight the differences between the standard deviation and the average of the differences with an example concerning the mathematics grades of six different twelfth grade classes (numbered from 1 to 6). Each class has 10 students.
The highest grade is 11, and the lowest is 1.
The average grade in each class is 6.
The distribution of the grades in each of the classes is displayed in the next slide.
For example, in twelfth grade class no. 1 (the first row):
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One child received a grade of 8.
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2 children received a grade of 7.
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4 children received a grade of 6.
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2 children received grades of 5.
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One child received a grade of 4.
Table 3.18
| grade | 11 pts. | 10 pts. | 9 pts. | 8 pts. | 7 pts. | 6 pts. | 5 pts. | 4 pts. | 3 pts. | 2 pts. | 1 pt. |
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) |
| Class no. 1 | ☺ | ☺☺ | ☺☺ ☺☺ |
☺☺ | ☺ |
||||||
| Class no. 2 | ☺☺ | ☺☺ | ☺☺ | ☺☺ | ☺☺ | ||||||
| Class no. 3 | ☺☺ | ☺☺ ☺ |
☺☺ ☺ |
☺☺ | |||||||
| Class no. 4 | ☺☺ ☺ |
☺☺ | ☺☺ | ☺☺ ☺☺ |
|||||||
| Class no. 5 | ☺☺ ☺☺ ☺ |
☺☺ ☺☺ ☺ |
|||||||||
| Class no. 6 | ☺☺ ☺☺ ☺ |
☺☺ ☺☺ ☺ |
|||||||||
| Calculations | |||||||||||
| grade | Average (M) | Average Difference | Standard deviation (σ) | ||||||||
| (1) | (13) | (14) | (15) | ||||||||
| Class no. 1 | 6 pts. | 0.8 pts. | 1.1 pts. | ||||||||
| Class no. 2 | 6 pts. | 1.6 pts. | 2 pts. | ||||||||
| Class no. 3 | 6 pts. | 2.4 pts. | 2.45 pts. | ||||||||
| Class no. 4 | 6 pts. | 3.6 pts. | 3.63 pts. | ||||||||
| Class no. 5 | 6 pts. | 4.0 pts. | 4.0 pts. | ||||||||
| Class no. 6 | 6 pts. | 5.0 pts. | 5.0 pts. | ||||||||
Symmetric Distribution
In the previous example, the distribution of marks on both sides of the average is symmetrical for each class, i.e. the number of children who received a mark higher than the average by any specific number of points is the same as the number of children who received a mark lower than the average by that same number of points.
When the distribution is very unsymmetrical, large differences between the standard deviation and the average difference are possible.
One child received a grade of 4.
Looking at the table, we see that in the lower rows, the grades are more widely-dispersed around the average grade than they are in the higher ones. This fact is obviously reflected in the calculation of the average difference (column 14) and the standard deviation (column 15).
Starting with row 5, column 14 is equal to column 15.
In the first 4 rows column 15 is higher than column 14, but the gap decreases in the lower rows.
The figures in columns no. 14 and 15 are measured in grade points.
For example, in row 3, the average difference (from M) is 2.4 points, and the standard deviation is 2.45 points.
