Mathematics
Variance
Quick fact
To compute variance, you first square each difference from the mean, then average those squares. This squaring makes larger deviations count disproportionately more.
Why this is interesting
You know that the average of your friends' heights tells you something about them, but does it tell you whether they are all nearly the same height or wildly different? Variance is the number that captures exactly that distinction.
Read the full explanation
Understanding Variance
Imagine you have the heights (in cm) of five friends: 150, 152, 151, 149, 153. The mean is 151. Now, most friends are close to the mean, so the spread is small. But if you had heights 140, 160, 150, 145, 170, the mean is still 151, yet the values are much more spread out. Variance measures this spread. For each data point, you subtract the mean to get a deviation: some are positive, some negative. If you simply average these deviations, they cancel out—you'd always get zero. So instead, you square each deviation (making them all positive) and then average those squares. The result is the variance: a single number that tells you how far, on average, the data points are from the mean, in squared units. A higher variance means more spread; a variance of zero means every number is identical.
A deeper explanation
Why does squaring make sense? It solves the cancellation problem, but it also gives extra weight to outliers—a point far from the mean contributes a disproportionately large squared deviation. This can be useful, but it also makes variance sensitive to extreme values. Mathematically, the variance is the average of the squared differences from the mean: for a population, σ² = (Σ(xᵢ − μ)²) / N. For a sample, we use n−1 in the denominator to correct for the fact that we are estimating from a subset, giving an unbiased estimate: s² = (Σ(xᵢ − x̄)²) / (n−1). Squaring also means the variance is in squared units, which is hard to interpret. That is why we typically take the square root to get the standard deviation, which is in the original units. Variance is central to probability theory: the variance of a random variable measures its expected squared deviation from the mean. It appears in the law of total variance, the Central Limit Theorem, and ANOVA.