Range, Variance, and Standard Deviation

Two data sets can share a mean and look nothing alike. A measure of spread is what tells them apart.

Range

The range is maximum minus minimum. It is quick, and it depends entirely on the two most extreme values.

Deviations

A deviation is a value minus the mean. Averaging them directly fails, because they always total zero.

Why squaring

Squaring removes the signs so the deviations stop cancelling. The average of those squares is the variance.

Back to real units

Variance is measured in squared units, which is unusable. Taking the square root gives the standard deviation, in the original units.

What it means

The standard deviation is roughly the typical distance from the mean. A large one means the values scatter widely.

Centre alone is not enough

Two data sets with identical means can be wildly different — one tightly clustered, one scattered. Reporting a centre without a spread hides exactly the variability that made the question statistical.

Standard deviation is a typical distance

It measures roughly how far observations sit from the mean. Squaring the deviations before averaging prevents positives and negatives cancelling, and the square root returns the answer to the original units.

Variance is the square

Variance is the standard deviation squared, so its units are squared too — square metres for heights, which is why standard deviation is usually reported instead. Variance survives because it adds nicely.

The range is the crude one

Maximum minus minimum uses only two values and is destroyed by a single outlier. It is easy to compute and tells you very little about the bulk of the data.