Why calculate the standard deviation and variance
When a standard deviation is known, then, if the scores are normally distributed, we know where certain scores will fall in relation to other scores, which can be useful…around 68% of the scores fall within one standard deviation [on] either side of the mean, and around 95% fall within two standard deviations [on] either side of the mean. If we know the mean and the standard deviation of a set of scores, we can tell where a particular score falls as a percentage of other scores.
For example, if on a memory test I recall 20 words out of 30, I may not know whether this is a good score or a bad score. However if I know that the mean of the scores of a whole group doing the test was15, and that the standard deviation was 2.5, then I know that my score of 20 is just within the top 16% of all the scores…Knowing the mean and the standard deviation researchers could work out whether a score is good or bad, because if the scores are normally distributed, the percentages of scores within one or two standard deviations remains the same in every set of [normally distributed] scores. If the normal distribution curve of a set of scores has a fairly flat curve, then the standard deviation is likely to be large (as around 68% of the scores will be quite widely spread around the mean). If the normal distribution curve of a set of scores has a tall curve, then the standard deviation is likely to be small (as around 68% of the scores will be clustered around the mean). Standard deviation measures the width of the distance that around 68% of the scores lies [on] either side of the mean (and two standard deviations gives around 95% [on] either side of the mean).
Another reason for calculating standard deviation and variance (the standard deviation squared) is that some inferential tests can be used only under certain conditions, one of which involves variance. To compare two sets of scores using some of the tests…the scores must be normally distributed and have [a] similar variance because some of the tests used involve using mathematical calculations such as squares and square roots.
Brain, Christine. Advanced Psychology: Applications, Issues and Perspectives. Cheltenham, Gloucester: Nelson Thornes, 2001, p.317.