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.