The Standard Deviation
The standard deviation is a measure that summarises the amount by which every value within a dataset varies from the mean. Effectively it indicates how tightly the values in the dataset are bunched around the mean value. It is the most robust and widely used measure of dispersion since, unlike the range and inter-quartile range, it takes into account every variable in the dataset. When the values in a dataset are pretty tightly bunched together the standard deviation is small. When the values are spread apart the standard deviation will be relatively large. The standard deviation is usually presented in conjunction with the mean and is measured in the same units.
In many datasets the values deviate from the mean value due to chance and such datasets are said to display a normal distribution. In a dataset with a normal distribution most of the values are clustered around the mean while relatively few values tend to be extremely high or extremely low. Many natural phenomena display a normal distribution.
For datasets that have a normal distribution the standard deviation can be used to determine the proportion of values that lie within a particular range of the mean value. For such distributions it is always the case that 68% of values are less than one standard deviation (1SD) away from the mean value, that 95% of values are less than two standard deviations (2SD) away from the mean and that 99% of values are less than three standard deviations (3SD) away from the mean. Figure 3 shows this concept in diagrammatical form.
If the mean of a dataset is 25 and its standard deviation is 1.6, then
- 68% of the values in the dataset will lie between MEAN-1SD (25-1.6=23.4) and MEAN+1SD (25+1.6=26.6)
- 99% of the values will lie between MEAN-3SD (25-4.8=20.2) and MEAN+3SD (25+4.8=29.8).
If the dataset had the same mean of 25 but a larger standard deviation (for example, 2.3) it would indicate that the values were more dispersed. The frequency distribution for a dispersed dataset would still show a normal distribution but when plotted on a graph the shape of the curve will be flatter as in figure 4.