Variance quantifies how far a collection of numbers spreads out from their arithmetic mean.
It calculates the expected value of the squared deviations of observations from their sample or population mean.
To compute sample variance, subtract the sample mean from each observation to determine individual deviation distances.
Next, square each deviation so that positive and negative deviations both contribute positively to the total sum.
Summing the un-squared deviations always equals zero, which is why raw differences cannot measure dispersion.
For a population of N items, the sum of squared deviations is divided by N to produce the population variance.
For a sample of n observations, the sum of squared deviations is divided by n minus one rather than n.
Dividing by n minus one is known as Bessel correction, which compensates for using the sample mean instead of the true population mean.
Without Bessel correction, the sample variance systematically underestimates the true population variance.
For example, for values two, four, and six with mean four, squared deviations are four, zero, and four, giving a sample variance of four.
Because deviations are squared, large departures from the mean receive disproportionately heavy mathematical penalties.
This quadratic penalty makes variance sensitive to extreme outliers that lie far from the central cluster.
The primary drawback of variance is its dimensional unit, which equals the square of the original measurement scale.
For instance, measuring patient weight in kilograms yields a variance expressed in squared kilograms, which lacks physical intuition.
Variance is additive for independent random variables, making it an indispensable theoretical tool across probability and statistics.