Trace [2,4,4,6] Mean=4; centered deviations [-2,0,0,2]. Squared deviations sum to 8. Population variance=2; sample variance=8/3. Standard deviation is the square root; its units match the observations. 2 / 6
Median preserves observations Sort a copy, never the paired data in place. Odd length: middle index n//2. Even length: average indices n//2-1 and n//2. 3 / 6
Trace paired correlation x=[1,2,3], y=[1,3,2]; both means=2. Pair-product sum=1; each squared-deviation sum=2. r=1/sqrt(2×2)=0.5. Independently sorting columns would invent r=1. 4 / 6
Boundary meanings Sample variance requires at least two observations. Constant-column correlation returns 0 by API policy; the ratio is undefined. Zero correlation does not rule out y=x². A normal-looking mean distribution does not make raw observations normal. 5 / 6
Transfer checks Add 100: variance unchanged. Multiply by -3: variance ×9, standard deviation ×3. Complete the [1,3,5] trace, then verify input preservation. 6 / 6