Variance and Standard Deviation

Variance and standard deviation are the same fact in two costumes. Variance averages the squared deviations from the mean, which makes it mathematically well behaved, variances of independent quantities simply add, but leaves it in squared units, so the variance of a set of heights in centimetres comes out in cm². Taking the square root undoes that and returns the standard deviation, a spread you can actually mark on the axis alongside the data. Ronald Fisher coined the word "variance" in a 1918 paper on Mendelian inheritance, precisely because he needed a quantity that partitioned additively across sources of variation; the whole of analysis of variance grew out of that choice.

The common trap is averaging standard deviations. You cannot: if two independent measurements have σ = 3 and σ = 4, the combined spread is √(9 + 16) = 5, not 3.5. Add the variances, then take the root. Numerically the conversion is trivial in both directions: a standard deviation of 12 means a variance of 144, and a reported variance of 6.25 means a standard deviation of 2.5.