Variance

The mean tells you the center. But two very different sets of numbers can share the same center. The returns 5, 5, 5 and the returns negative 20, 5, 30 both average to about the same place, yet one is perfectly steady and the other swings wildly. To capture that difference, we need a measure of spread, of how far the values tend to stray from the mean. Variance is that measure, and this lesson takes its formula apart piece by piece so it makes sense.

Start with the deviations from the mean

Spread is about distance from the mean, so the natural first step is to find each value's deviation, meaning how far it sits from the mean. Take the numbers 2, 4, 6, and 8, whose mean is 5. The deviations are 2 minus 5 which is negative 3, 4 minus 5 which is negative 1, 6 minus 5 which is 1, and 8 minus 5 which is 3. Values below the mean give negative deviations, and values above give positive ones.

Why you cannot just add the deviations

Here is the first puzzle. It seems natural to just average the deviations to get a typical distance from the mean. But watch what happens when you add them: negative 3 plus negative 1 plus 1 plus 3 equals zero. This is not a coincidence. The deviations always sum to exactly zero, for any dataset, because the mean is the balancing point. The negatives below the mean perfectly cancel the positives above it. So the plain average of the deviations is always zero and tells you nothing about spread. We need a way to stop the cancellation.

Why we square the deviations

The fix is to square each deviation before averaging. Squaring does two important jobs at once. First, it makes every deviation positive, since a negative times a negative is positive, so the below-mean and above-mean distances no longer cancel. Second, it penalizes large deviations far more than small ones. A deviation of 3 becomes 9, but a deviation of 6 becomes 36, four times as much even though the distance only doubled. This matches how we think about risk, because a value very far from the mean is more alarming than one slightly off. Variance is the average of these squared deviations, and it is finally a number that grows with spread instead of always collapsing to zero.

The formula in symbols

In plain words: take each value, subtract the mean, square the result, add up all those squares, and divide by the number of values. The worked example below walks through those exact steps with real numbers.

The drawback of squared units

Squaring solves the cancellation problem but creates a new inconvenience. Because we squared the deviations, the variance is in squared units. If your returns are in percent, the variance is in percent squared, which nobody can interpret intuitively. What does 5 percent squared even feel like? This awkwardness is exactly why the next lesson introduces the standard deviation, which simply takes the square root of the variance to bring the measure back to normal units. Variance and standard deviation are two views of the same idea of spread, one in squared units and one in the original units.

Variance in finance

In finance, variance measures how spread out an asset's returns are around their average, which is the raw statistical definition of risk. A stock whose returns cluster tightly around the mean has low variance and feels steady. A stock whose returns swing far above and below the mean has high variance and feels risky. Variance is also the quantity that combines across assets when you build a portfolio, which is why the portfolio variance lesson later in this unit depends directly on it. For now, the key is that variance is the average squared distance from the mean, and squaring is what makes it work.

Variance measures spread, but in awkward squared units. The next lesson takes its square root to get the standard deviation, the measure of spread in normal units and the number finance calls volatility.