In more detail
variance = Σ(x − mean)² / n for a whole population, or divided by n − 1 for a sample (Bessel's correction). Squaring removes signs and weights large misses more. Its units are the square of the data's units (metres become square metres), which is why the Standard deviation (its square root) is used for reporting. For the data 2, 4, 4, 4, 5, 5, 7, 9 the population variance is 4 and the sample variance 32 / 7 = 4.571.
Why it matters
Variance has algebra that standard deviation lacks. Adding a constant to every value leaves it unchanged; multiplying every value by a leaves it multiplied by a², so doubling the data quadruples the variance. For independent quantities, variances add, but standard deviations do not. This is how errors combine, how portfolio risk is computed and why averaging n independent measurements divides the variance by n.
Example
Two independent timing errors with standard deviations 3 ms and 4 ms:
combined variance = 3² + 4² = 9 + 16 = 25
combined SD = √25 = 5 ms (not 3 + 4 = 7 ms)
Converting a variance of 4 cm² to millimetres multiplies it by 10² = 100, giving 400 mm², while the standard deviation goes from 2 cm to 20 mm.
Common confusion
The Standard deviation is in the data's own units and is the figure to quote; the variance is the one to add up and to use inside formulas.