Spread Does Not Travel the Way Center Does

You'll learnhow variance weights squared distances by probability and why only the standard deviation carries the variable's units, what shifting and scaling each do to center and spread, why variances add for independent sums AND differences while standard deviations combine like perpendicular legs, and why doubling one observation is more variable than adding two independent ones.

Shift every value of a random variable by 3 and the mean moves by 3 while the spread does not move at all — the gaps between values are untouched. Double every value and both move. Combine two independent variables and the means add or subtract as expected, while the variances add either way: a difference of two uncertain quantities is more uncertain than either one. This page does not ask you to take any of that on faith. Every rule is applied to an actual list and re-computed from the definition — the shifted list, the doubled list, all four cells of a joint distribution — and the two roads agree to fourteen decimal places, on every load. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · VARIANCE TWO WAYS · SHIFT VS SCALE · THE PYTHAGOREAN COMBINE · 2X VS X1 + X2 · v2

Spread survives a shift, doubles under a scale, and combines like legs

Before you start

What you're looking at

The pets distribution again — five values with five probabilities — now with a violet fulcrum at its mean and a band one standard deviation wide either side.

The question

What happens to that spread when you shift the variable, stretch it, or add a second one to it?

Step 1 / 12