SUMMARISING AND PRESENTING DATA

MEASURES OF VARIABILITY

Standard Deviation and Variance

The standard deviation describes the "average distance" of data values from their mean.

The distance of each value x_i from the mean is

The mean of those distances, is always zero since

Instead we could use the squares of the distances (because the square of a negative number is positive) i.e.

But there is still a problem with the squared distances have units squared (e.g. if x_i are lengths, are lengths squared or areas)

So we take the square root, namely

Usually n-1 instead of n is used in the denominator because this gives an estimate with slightly better mathematical properties.

i.e. use s =

This is called the sample standard deviation.

The value obtained before taking the square root is called the sample variance. It is denoted by .

Example - Data set A

There are n=7 observations and = 4.57. The deviations from the mean, d_i = x_i - , are:   -2.57, -1.57, -1.57, -0.57, 0.43, 2.43, 3.43. So