Variance
back to Statistical Dispersion
The variance of a set of data is the mean of the sum of squared deviations from the Arithmetic Mean of the same set of data. Because this calculation sums the squared deviations, the unit of variance is the square of the unit of observation. Thus, the variance of a set of heights measured in inches will be given in square inches. This fact is inconvenient and has motivated statisticians to call the square root of the variance, the Standard Deviation and to quote this value as a summary of dispersion.
*Variance = {(x1-m)2 + (x2-m)2 + ... + (xn-m)2} / n; where m is the mean
When the set of data is a Population, we call this the population variance. If the set is a Sample, we call it the sample variance.
The method of calculation may be more easily understood from the table below where the mean is 8.
i x[i] x[i]-mean (x[i]-mean)^2
(deviation) (squared deviation)
1 5 -3 9
2 7 -1 1
3 8 0 0
4 10 2 4
5 10 2 4
--- ---- --- ---
n=5 sum=40 0 18
mean = 40/5 = 8
variance = 18/5 = 3.6
standard deviation = 1.897366596101 or 1.9
Note that the column of deviations sums to zero. This is always the case. Note also that we round the standard deviation to one more than the number of significant digits in the mean.