Centre Is Only Half the Story
Two towns both average 15 °C. One never leaves 12 to 18. The other swings from −20 to +45. Same centre, opposite realities — and spread is the whole difference.
Eleven Commutes
Ten ordinary mornings and one bad one. They sum to 341, so x̄ = 31 minutes and the median is 29. Every measure ahead is judged by how it reacts to that 53.
The Range
53 − 21 = 32 minutes. Honest and instant — but nine of the eleven values took no part, and the range only ever grows as you collect more days.
Variance
Deviations from the mean always sum to exactly zero, so square them first. The squares total 804, giving s² = 804 ÷ 10 = 80.4.
Standard Deviation
A variance of 80.4 is in square minutes. Take the root and the units return: s = 8.97 minutes. A typical morning differs from the average by about nine minutes.
Why n − 1?
x̄ is the centre that makes the sum of squares as small as it can be, so spread measured around it always comes out too small. Dividing by n − 1 puts back exactly what was lost.
Quartiles and the IQR
Q₁ is the median of the lower half (24), Q₃ of the upper half (35). Their gap is the middle 50% of the data: IQR = 11 minutes. Change the 53 to 200 and it does not move.
The Five-Number Summary
Minimum, Q₁, median, Q₃, maximum. Read the gaps and the shape appears without a chart: 3, then 5, then 6, then 18 — widening sharply to the right.
Building a Box Plot
- Box — Q₁ 24 to Q₃ 35, so its width is the IQR
- Median line — at 29, off-centre because the data is skewed
- Whiskers — to the last value inside the fences 7.5 and 51.5: so 21 and 38
- Lone points — anything past a fence, drawn on its own: the 53
Drag the Worst Commute
Coefficient of Variation
8.97 ÷ 31 = 28.9%. Record heights in metres instead of centimetres and s shrinks a hundredfold — the CV stays 4.1%. That is what lets you compare a commute with body temperature's 1.1%.
Spotting Outliers
The fence flags our 53. The z-score does not: z = (53 − 31) ÷ 8.97 = 2.45. Set the 53 aside and it scores 4.41 — it inflated the very s it was judged against. That is masking.
What you learned
Range is a sanity check. s = 8.97 uses every value and pays for it in fragility. IQR = 11 is robust. The five-number summary draws the box plot. CV = 28.9% is unitless. And an outlier can hide inside its own z-score.