Why Do We Square the Differences When Calculating Variance?
Ask why variance squares its deviations and many people assume it is just a rule to memorise. It is not. Squaring solves a specific problem that would otherwise make the calculation useless.
The Problem: Deviations Cancel Out
The mean is the balance point of the data, so the deviations from it always add up to exactly zero, however spread out the data is.
Worked Example
Take the data 2, 6, 10.
- The mean is 18 / 3 = 6.
- Deviations: −4, 0, 4. Their total is 0.
Averaging the raw deviations gives 0 for this data, and for every other data set too. It cannot tell a tightly clustered data set from a widely scattered one.
The Fix: Square Them
Squaring a non-zero number always gives a positive result (and 0 stays 0), so a value 4 below the mean and a value 4 above it both add 16 instead of cancelling.
- Squared deviations: 16, 0, 16. The sum of squares is 32.
- Population variance: 32 / 3 ≈ 10.667. Sample variance: 32 / 2 = 16.
Larger Deviations Count More
Squaring does not treat all deviations equally. A deviation twice as large contributes four times as much; three times as large, nine times as much. Variance is therefore especially sensitive to values far from the mean.
What About Absolute Values?
Taking absolute values also removes the minus signs. For the same data the absolute deviations are 4, 0, 4. They add up to 8, and dividing by 3 gives about 2.667. This is the mean absolute deviation, a perfectly valid measure of spread. It is less affected by outliers than variance.
| Approach | Result for 2, 6, 10 |
|---|---|
| Average raw deviation | 0 (always) |
| Average squared deviation (population variance) | ≈ 10.667 |
| Average absolute deviation | ≈ 2.667 |
Squares are used more widely because they behave well mathematically: variances of independent quantities add, squares can be differentiated (which least-squares regression relies on), and the normal distribution is described naturally by its variance.
Squaring and the Standard Deviation
Squaring changes the units: dollars become dollars squared. Taking the square root of the variance reverses that, giving the standard deviation in the original units. For the example above the population standard deviation is √10.667 ≈ 3.266.
The Cost: Sensitivity to Outliers
Because large deviations are amplified, one extreme value can dominate the variance. That is a feature when large errors really are worse than small ones, and a limitation when the outlier is just a recording mistake. Check unusual values before trusting the result.
Summary
Raw deviations always cancel to zero; squaring stops that, so every value adds to the total. The side effects (more weight on large deviations and squared units) explain why variance is sensitive to outliers and why we usually report its square root, the standard deviation.