The formula
s² = Σ(x − x̄)² / (n − 1)
What it means
Variance is the average of the squared differences from the mean. Squaring stops the differences above and below cancelling each other out, and the price is that the answer ends up in squared units: the variance of heights in centimetres comes out in square centimetres, which means nothing physically. That is why reports almost always show its square root, the standard deviation, and variance stays as an intermediate step.
How to work it out by hand
- Work out the mean of the values
- Subtract the mean from each value and square the difference
- Add all the squares: that is the sum of squares
- Divide by n − 1 for the sample version or by n for the population one
What is worth knowing
Variance has a property that standard deviation does not, and it is why it is still used: the variance of a sum of independent variables is the sum of their variances. That does not hold for standard deviations, which have to be converted to variances and back. Hence its appearance in nearly every error-propagation and portfolio formula, even though the figure finally published is always the deviation.