How Far Things Spread
Range, interquartile range, and standard deviation measure how far data sits from its center, and they differ chiefly in how much an outlier can move them. · 12 min
Two classes both average 70 on a test. In the first, every score lands between 68 and 72. In the second, scores run from 40 to 100. Same center, utterly different classes — and the mean cannot tell them apart. What separates them is spread: how far the values sit from the center. There are three common rulers for it. They agree on the tidy cases and disagree, revealingly, whenever a single value strays far from the rest.
Guess before you learn
You add one wildly large outlier to a dataset. Which of these three measures of spread will change the least?
The IQR. It looks only at the middle half of the data, so a single extreme value — which lands out at the very edge — cannot reach the quartiles that define it. The range jumps immediately, and the standard deviation moves a lot too. If you guessed one of those, keep the mark: this folio is about exactly which rulers an outlier can bend.
9–12
3–5
Spread means: are the numbers close together or far apart? The easiest measure is the range — the biggest value minus the smallest. A small range means everyone is bunched up; a big range means they are scattered.
The trouble is that the range only looks at the two ends. One odd value at an end can make a bunched-up group look scattered. Better rulers look at the whole group, not just the edges.
6–8
Three rulers measure spread. The range is maximum minus minimum. The interquartile range (IQR) is the width of the middle 50%: sort the data, find the value a quarter of the way up (Q1) and three-quarters up (Q3), and subtract. The standard deviation is roughly the typical distance of a value from the mean.
They differ in how much of the data they use. The range uses only the two extremes, so one outlier controls it entirely. The IQR ignores the outer quarters, so it shrugs off outliers. The standard deviation uses every value, so an outlier moves it — a lot, because of how it is built.
9–12
The range = max − min uses only two values, so a single outlier sets it. The IQR = Q3 − Q1 is the span of the middle half of the sorted data; because it discards the outer quarters, no lone extreme can touch it. The standard deviation s = √( Σ(x − x̄)² / (n − 1) ) is the root-mean-square distance from the mean — the typical gap, in the data's own units.
The standard deviation squares every deviation before averaging, so distance from the mean counts far more than proportionally: a value ten units out contributes a hundred, one twenty units out contributes four hundred. That squaring is what makes the standard deviation sensitive to outliers and heavy tails. The pairing is not arbitrary: report the mean with the standard deviation, the median with the IQR — a resistant center deserves a resistant spread.
K–2
Line up your friends by height. The range is the gap from the shortest to the tallest — one big jump across the whole line.
But one very tall visitor makes that gap huge, even if everyone else is close together. So the range can be fooled by a single unusual person.
Undergrad
The three statistics differ in breakdown point, the fraction of contamination they tolerate before diverging: 0 for the range and the standard deviation, 1/4 for the IQR, whose outer quarters can be arbitrarily corrupted while Q1 and Q3 hold. The n − 1 divisor (Bessel's correction) makes s² unbiased for the population variance: dividing by n underestimates, because deviations are taken about the sample mean, which sits closer to the data than the true mean.
Robust scale estimators sharpen the IQR's idea. The median absolute deviation, MAD = median|x − median(x)|, has the maximal breakdown point 1/2; scaled by 1.4826 it estimates σ at the normal model. Choosing a spread is again a bet about contamination: a few percent of outliers make the standard deviation unreliable, while the IQR and MAD barely flinch. Reporting s with x̄ and the IQR with the median keeps center and spread equally resistant.
Postgrad
As functionals, the standard deviation has an unbounded influence function, quadratic in the deviation, so one observation exerts arbitrary leverage — the analytic form of its zero breakdown point. Quantile-based spreads have bounded influence: the IQR's depends on the density only at the quartiles. This is the same robustness ledger seen for center, now for the second moment, and it is why exploratory practice pairs medians with hinges (Tukey's boxplot) rather than moments.
Efficiency exacts its price in reverse. At the normal model the IQR and MAD are far less efficient than s — the MAD's asymptotic efficiency is about 37% — so robustness costs precision on clean data, as it did for the median. Modern scale estimation (Rousseeuw and Croux's Qn, Winsorised variances, M-estimators) seeks the frontier: high breakdown with efficiency near s. Reporting a spread, like a center, declares a model of how the data may be dirty.
interquartile range (IQR)
The width of the middle 50% of sorted data: IQR = Q3 − Q1, where Q1 sits a quarter of the way up and Q3 three-quarters up. It is resistant — a single extreme value cannot change it.
Find the interquartile range of 4, 7, 8, 9, 11, 12, 20 — the steps fade as you master them
4, 7, 8, [9], 11, 12, 20 → median = 9
lower half 4, 7, 8 → Q1 = 7
upper half 11, 12, 20 → Q3 = 12
12 − 7 = 5
Now push one value away and watch the three rulers respond. Start with a calm dataset. Drag a single point steadily outward until it is a far outlier. The range tracks it step for step. The standard deviation, which squares every distance, climbs almost as fast. The IQR — anchored in the middle half — does not move at all. Sketch the standard deviation's climb before the ink reveals it.
Why is this true?
Why can a single enormous outlier leave the IQR completely unchanged?
Because the IQR is built from Q1 and Q3, the values a quarter and three-quarters of the way through the sorted data. A lone extreme value sits out past everyone, in the top or bottom quarter, so it never becomes a quartile. It changes who the maximum is, not who stands at the one-quarter and three-quarter marks.
You can now describe a single variable in full: its level, its center, and its spread. But three numbers still flatten a distribution into a summary. Next folio restores the picture — the shape of the data — and shows that the shape itself is what tells you whether to trust the mean or the median in the first place.
Practice — new ink and old, interleaved
1.Every value in a dataset is 7: the data are 7, 7, 7, 7. What is the standard deviation?
2.A report on wait times wants a spread figure that a handful of very long waits will not exaggerate. Which should it use?
3.For the sorted data 2, 4, 4, 6, 8, 10, 12, 14 (eight values), Q1 = 4 and Q3 = 11. What is the IQR?
4.Of the four levels — nominal, ordinal, interval, ratio — on how many is the mean a legitimate summary of center?
5.Computing a standard deviation means adding and squaring differences between values. On which level of measurement does it first become legal?
6.Find the mean of 12, 15, 18, 21, 24.
7.Find the median of 5, 2, 9, 4, 12, 7.
8.For a dataset the mean is 71 and the median is 52. What does this say about the shape?
9.The same right-skewed wait times have a mean of 22 minutes and a median of 14. Which is the more honest headline figure for a typical wait?