The R Book

Variance

A measure of variability is perhaps the most important quantity in statistical analysis. The greater the variability in the data, the greater will be our uncertainty in the values of parameters estimated from the data, and the less will be our ability to distinguish between competing hypotheses about the data.

The variance of a sample is measured as a function of ‘the sum of the squares of the difference between the data and the arithmetic mean’. This important quantity is called the ‘sum of squares’:

Naturally, this quantity gets bigger with every new data point you add to the sample. An obvious way to compensate for this is to measure variability as the average of the squared departures from the mean (the ‘mean square deviation’.). There is a slight problem, however. Look at the formula for the sum of squares, SS, above and ask yourself what you need to know before you can calculate it. You have the data, y, but the only way you can know the sample mean, , is to calculate it from the data (you will never know in advance).