Unit 6: Business Decision Making - Techniques to analyse data
Standard deviation and dispersion
What is dispersion?
- how data is dispersed.
- how data is spread over the whole range of data.
- mean, median and mode give an indication of central tendency, dispersion gives an indication of spread.
- measures of spread include: range; mean absolute deviation, variance and standard deviation
Range
The range is the difference between the highest data value and the lowest data value.
With a data range such as: 95, 52, 13, 45, 59, 23, 37, the range is 95-13 = 82
The range does not tell us how the data is spread, but does tell us the extent of the spread.
Standard deviation
In order to caulate the standard deviation of any data sample we first need to understand the mean absolute deviation and the variance.
Mean Absolute Deviation
Here we measure the average deviation from the mean.
Consider a very simple set of numbers
1, 2, 3, 4, 5, the mean of this set of numbers is 3
Tablulating this data into a graph we measure the deviation of each entry from the mean. The absolute deviat
Number |
Deviation from the Mean |
Absolute Deviation |
1 |
3 - 1 = 2 |
2 |
2 |
3 - 2 = 1 |
1 |
3 |
3 - 3 = 0 |
0 |
4 |
3 - 4 = -1 |
1 |
5 |
3 - 5 = -2 |
2 |
|
Total |
6 |
The sum of the absolute deviations is 6
The total absolute deviation from the mean is 6
The average absolute deviation from the mean is 6 + 5 = 1.2
The mean absolute deviation (MAD for short) from the mean is 1.2
The variance
We can overcome the problems of "negative deviations" by squaring the deviations, because when two negative numbers are multiplied together the result is a postive number.
Number |
Deviation from the Mean |
Squared Deviation |
1 |
3 - 1 = 2 |
4 |
2 |
3 - 2 = 1 |
1 |
3 |
3 - 3 = 0 |
0 |
4 |
3 - 4 = -1 |
1 |
5 |
3 - 5 = -2 |
4 |
|
Total |
10 |
The sum of the squared deviations is 10
The total squared deviation from the mean is 10
The average squared deviation from the mean is 10 + 5 = 2
The variance from the mean is 2
The only problem now is units, if it was cm, the variance is now cm²
Standard deviation
The standard deviation is the square root of the variance, as the variance is the sum of the squares of differences, then taking the square root of that sum is logical
Standard deviation is the most commonly used measure of dispersion
Using the above example, the standard deviation is the square root of 2 which is equal to 1.41 (two decimal places).