Measuring dispersion

• The standard deviation is a statistic that tells you something about how wide the dispersion of the data is from the mean. To calculate it, first find the mean. Then take each value in the set and subtract the mean from it. Square each difference, add up all the squares and divide the result by the number of values minus 1. Now you’ve got the variance. But the variance is expressed in square units of the data, and that isn’t always useful. For example, the variance of the pocket money sample is 107.52 square pounds, but there’s no such thing as a ‘square pound’. Take the square root of this number to get the standard deviation: £10.37. Even if you knew nothing else about the data, the fact that the mean and standard deviation are close would tell you that the data have a fairly wide dispersion.

  What about the different dispersions of the two groups? A glance at the boxplot tells you the range of values isn’t the same between 13 and 14 year olds. By dividing the standard deviation by the mean, we get the coefficient of variation. This is a comparable measure of variation within distinct groups. For the 13-year-olds, the standard deviation is 12.27 and the mean is 12.13, giving a coefficient of variation of 1.01. For the 14-year-olds, the standard deviation is 7.42 and the mean is 11.59, giving a coefficient of variation of 0.64. The 13-year-olds therefore have a wider variation of pocket-money values.

  Excel and SPSS can calculate variance and standard deviation for you, so you don’t have to perform all these calculations manually. However, they won’t calculate the coefficient of variation for you, so just divide the standard deviation by the mean yourself.