Measures of Dispersion: Range and Standard Deviation

The Range

The range is a simple measure of dispersion. You subtract the lowest score in the data set from the highest score to give the range.  The range is a very simplistic measure and does not use all the scores in the data set therefore it can be distorted by a very high or low score that does not reflect the range of most of the other scores in between those two points.

Standard deviation and when to use it

  • When you want to know how spread out a set of scores (data-points) are around the mean and the level of measurement is interval/ratio, the standard deviation is appropriate.

How do you work it out?

If you are allowed to you could use a handy online calculator like this one:

If you are in an exam, you may need to be able to calculate the SD for yourself using the formula below, which will be shown at the beginning of your exam booklet (Edexcel A Level).

  • File_000In an exam, the question might have a table for you to complete, if not then you should be very careful and lay out your workings neatly – you will lose marks unless you write out each stage accurately
  • So first off remember X with a line over it means “the mean” for this set of data.
  • So you need to start with the bit in brackets, neatly write out a column of figures where you calculate each score minus the mean, these are called d scores, d for deviation ,meaning how much does each score differ from the mean.
  • Some of these scores will be negative, because the score is smaller then the mean, some will be positive, because the score was bigger than the mean
  • To make all the d scores positive you need to square each of them, that is you need to calculate dxd and write it in the next column
  • you then need to remind yourself that the sigma symbol means sum total, so add up all the d squared scores
  • Now divide this number by the number of scores minus 1.
  • So this gives you a figure called the variance, but we want the standard deviation so there is one last things to do, square root this figure to arrive at your destination, the standard deviation!!

mindful standard deviation

Strengths of the standard deviation

  • When doing the calculation, we use every score and therefore is the most sensitive measure of central tendency telling us how ALL the data spread around the mean;
  • for example the range only uses the highest and lowest scores and therefore we know little about the majority of the scores and how close they were to the mean
  • if we want to use a parametric test (a set of more sensitive inferential stats tests, meaning that they help us to better avoid making type 1 and 2 errors) then we must have …
    • homogeneity of variance, meaning that the standard deviation needs to be similar in each group
    • normally distributed data (we need to know the standard deviation to work this out)
  • if these criteria are not met then we must stick to a non-parametric test, e.g. Spearman’s, Wilcoxon’s , Mann Whitney, Chi Squared

Weaknesses of using the standard deviation

  • if there are any outliers, that is extreme scores which do not reflect the majority of the data, at either end (high or low) then the standard deviation can be affected
  • standard deviation takes longer to calculate when done by hand

Timed practice question