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.

8 STEPS · 6 QUICK CHECKS · VARIANCE TWO WAYS · SHIFT VS SCALE · THE PYTHAGOREAN COMBINE · 2X VS X1 + X2 · v1

a bracket that shifts without stretching · legs at right angles · 200 against 141.4

Before you start

What you're looking at

The pets distribution from Topic 2.8 — mean 1.25 — about to get its second parameter: a standard deviation of 1.13 pets, computed term by term.

The question

"What happens to center and spread when a variable is shifted, scaled, or combined with another?" — answered by transforming the actual list and recomputing, not by quoting rules.

Watch for

The sigma bracket. It ignores a shift, doubles under a scale factor of 2, and combines with another bracket like perpendicular legs — never end to end.

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