Simple Statistics

The Standard Deviation
When we calculated the mean deviation (above), we did not take into account the fact that some numbers are higher than the mean and some are lower, i.e. our true deviations should have been –2, -1, 0, +1, +2.  However, if we add these together, our total would have be zero, so we ignored the signs for this reason.  Ignoring signs is not especially pleasing to mathematicians and partly for this reason, but mainly because the mean deviation is a very simple figure without any powerful mathematical properties, it is rarely used as a measure of dispersions. The preferred measure, known as the standard deviation is used far more often, usually in conjunction with the mean and the range.

In principle, the standard deviation (often shortened to ‘sd’) is very similar to the mean deviation.  It summarises an average distance of all the scores from the mean of a particular set. But it is calculated in a slightly different manner.
As just discussed, if we take into account the signs (+ or -) of the deviations from the mean, the mean deviation will always be zero.  There is however a solution to the problem.  If you multiply two negative numbers together, you get a positive result.  The same applies to squaring a negative number (multiplying the number by itself).

Now suppose you have the set of deviations –2, -1, 0, +1, +2; these equal zero when added together, or 6 when the signs are ignored.  If instead of adding the deviations, you square each one you get +4, +1, 0, +1, +4 for the squared deviations, and all are positive numbers. Magic!  Now you need to add these numbers and find their mean, just as you did with the mean deviation calculations.  You should get 10, divided by 5, to obtain the value 2.  The figure 10 is called the sum of squares, and 2, the mean of the sum of squares is called the variance. Notice that 2 is not the standard deviation.  As we squared all the differences, the figure 2 must be ‘unsquared’ (find its square root), to bring it back in perspective.  In this case, the square root of 2 is 1.4142.  This then is the standard deviation.  N.B. this is slightly higher than the figure 1.2 which was calculated as the mean deviation for the same set of numbers.

A slight (but not serious) complication with the standard deviation is the use of N (the number of scores in the particular set).  The majority of quantitative Market Research methodologies involve taking a sample from a population (as described above), however because this is only a sample, some error will inevitably occur in the results.  The sensible way to compensate for this is to make an allowance for it in the calculation of the standard deviation.  Some statistical books tell us to divide the sum of squares by N-1 instead of N i.e. dividing by a smaller number will give a larger variance, and hence a larger standard deviation, which compensates for the margin of error in taking a sample. Thus a formula for the standard deviation involving N-1 rather than N is preferred whenever we are working with samples rather than a set of scores which is absolutely complete (a population).