One number says where the data sits — a second says how far it scatters.
Ask two classes their test scores and both might average 80. Are the classes the same? Not at all — one might run a tight band from 76 to 84, the other swing from 62 to 98. The center we met in Lesson 34.3 pins down where the data sits; it says nothing about how widely it scatters. This lesson builds the second half of every honest summary — the spread. We will cut the data into quarters with quartiles, draw the five-number summary as a box plot, flag stragglers with the 1.5·IQR rule, and finally average the squared distances from the mean to get the variance and its square root, the standard deviation — the single number statisticians trust most for spread.
"You scored in the 90th percentile." It sounds like a grade, but it is a rank: you beat about 90% of the people who took the test. A percentile answers the question "what fraction of the data falls at or below this value?" The p-th percentile is the value with about p% of the data at or below it.
To find it we need a rule, because data comes in lumps. Sort the n values from smallest to largest, then find the position of the percentile:
Sort the data. The p-th percentile sits at position i = n · p%. If i is not a whole number, round it up to the next position and read that value. If i lands exactly on a whole number, average the two values it sits between (positions i and i+1).
Different books round slightly differently; what matters is that you declare your method and stay with it. This is the rule the figures and readouts on this page use, so the box never lies.
Sorted (minutes): 3, 7, 8, 12, 14, 15, 18, 21, 22, 25 (n = 10).
Position of P90: i = 10 · 0.90 = 9, a whole number → average positions 9 and 10: P90 = (22 + 25)/2 = 23.5 minutes. About 90% of the class finished in 23.5 minutes or less.
Position of P25: i = 10 · 0.25 = 2.5, not whole → round up to position 3: P25 = 8 minutes.
GX.percentile — read
the rank story underneath.Three special percentiles slice the sorted data into four equal-sized groups. They are the quartiles:
Q₁ = P25 — a quarter of the data is at or below it (the lower quartile).
Q₂ = P50 = the median — the halfway value.
Q₃ = P75 — three quarters of the data is at or below it (the upper quartile).
The distance between the outer two is the heart of spread:
IQR = Q₃ − Q₁ (the interquartile range)
The IQR is the width of the middle half of the data. Because it throws away the bottom quarter and the top quarter, it does not even notice a wild value at either end — it is the robust measure of spread, the spread to quote whenever outliers lurk, just as the median was the robust center in 34.3.
Sorted: 62, 71, 74, 77, 79, 82, 85, 88, 91, 95 (n = 10).
Q₁ = P25: i = 10·0.25 = 2.5 → up to position 3 → Q₁ = 74.
Q₂ = median: average the 5th and 6th values → (79 + 82)/2 = 80.5.
Q₃ = P75: i = 10·0.75 = 7.5 → up to position 8 → Q₃ = 88.
So IQR = Q₃ − Q₁ = 88 − 74 = 14. The middle half of the class is packed into a 14-point band.
Stack five numbers and you can draw a distribution. The five-number summary — min, Q₁, median, Q₃, max — becomes a box-and-whisker plot: a box from Q₁ to Q₃ (so the box is the middle half, its length the IQR), a line inside at the median, and whiskers reaching out to the smallest and largest ordinary values.
"Ordinary," because the box plot also flags stragglers. Build a fence one and a half IQRs beyond each quartile:
Lower fence = Q₁ − 1.5·IQR · Upper fence = Q₃ + 1.5·IQR.
Any value outside the fences is an outlier. It is drawn as its own red dot, and the whisker stops at the most extreme value that is still inside the fence — the whisker no longer chases the straggler.
GX.fiveNum.Notice what the stray value does: it barely nudges the median and IQR (the box hardly moves), yet it stretches the range dramatically. That contrast is exactly what the rest of this lesson sharpens.
The simplest spread of all just subtracts the ends:
range = max − min
The range is quick and honest about the full reach of the data, but it leans entirely on the two most extreme values — so a single outlier can blow it up. For our 10 exam scores the range is 95 − 62 = 33; add one stray 30 and the range leaps to 95 − 30 = 65, even though almost nothing about the class changed. The IQR over the same change went only from 14 to 17. Range and IQR answer different questions: range asks "how far apart are the extremes?", IQR asks "how wide is the typical middle?"
Class A scores: 76, 78, 80, 80, 82, 84 — mean 80, range 8.
Class B scores: 68, 74, 80, 80, 86, 92 — mean 80, range 24.
Identical centers, very different lives. We need a measure of spread that uses every value, not just the two ends — that is the variance, next.
To use all the data, measure how far each value sits from the mean and average those distances. But a plain average of (xᵢ − x̄) always cancels to zero — the mean is the balance point, so the positives and negatives offset exactly. The fix is to square each deviation first. Squaring kills the signs and, as a bonus, punishes far-away values more. Average the squared deviations and you have the variance:
s² = Σ(xᵢ − x̄)²n
The variance is in squared units (points², centimetres²) — awkward to talk about. So take the square root and return to the data's own units. That is the standard deviation:
s = √s² = √( Σ(xᵢ − x̄)²n )
Read s as "the typical distance of a value from the mean." Small s means the data huddles near x̄; large s means it sprawls.
Data: 2, 4, 4, 4, 5, 5, 7, 9. First the mean: x̄ = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5.
Deviations from 5: −3, −1, −1, −1, 0, 0, 2, 4.
Squared: 9, 1, 1, 1, 0, 0, 4, 16; their sum is 32.
s² = 32 / 8 = 4 (points²), s = √4 = 2. The values typically sit 2 away from the mean of 5.
The variance uses squared deviations. Two mistakes bite here: forgetting to square (averaging plain deviations gives 0 every time), and forgetting to divide by n (the sum of squares alone keeps growing as you add data). And keep range and IQR straight — an outlier inflates the range and s far more than it moves the IQR, because IQR ignores the extremes while s squares them.
A note for later: dividing by n gives the population variance σ². When the data is a sample used to estimate a larger population, statisticians divide by n − 1 instead (the "sample variance"); we meet that fix in Lesson 34.5. On this page we divide by n and say "population" so every number is exact.
Here is where spread earns its keep. Two archers, two basketball shooters, two suppliers, two investments — whenever the averages tie, the one with the smaller standard deviation is the steadier, more dependable one. Same mean, smaller s ⇒ the values cluster tighter, the surprises are smaller, the performance is more predictable.
GX.mean and
GX.dataSd.The conclusion is the one-liner to carry forward: a center tells you what to expect; the spread tells you how much to trust it. A summary that gives both — say, "mean 80, s = 2.6" — describes a class far better than either number alone.
| Measure | Formula | What it captures | Robust to outliers? |
|---|---|---|---|
| Range | max − min | full reach of the data | no |
| IQR | Q₃ − Q₁ | width of the middle half | yes |
| Variance s² | Σ(xᵢ−x̄)² / n | mean squared distance from x̄ | no |
| Std. deviation s | √s² | typical distance from x̄, in data units | no |
And the spine of the lesson: a p-th percentile marks the value with p% at or below it; Q₁, Q₂, Q₃ = P25, P50, P75 cut the data into quarters; the five-number summary draws a box plot and the 1.5·IQR fences flag outliers; and when centers tie, the smaller standard deviation wins for steadiness. Next, Lesson 34.5 uses a sample's x̄ and s to infer the whole population — and meets the bell curve where spread becomes σ.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops measures of spread alongside the five-number summary and box plots, addressing CCSS HSS-ID.A.1 (represent data with plots on the real number line — here, box plots), HSS-ID.A.2 (compare center and spread — IQR and standard deviation — of two or more data sets), and HSS-ID.A.3 (interpret differences in spread accounting for the effect of outliers). The quartile position method (i = n·p%, round up, average on a tie) is stated explicitly and used consistently; popular software packages use slightly different interpolation rules, so a student's hand answer may differ from a calculator by a small amount — the important habit is declaring the method. We divide squared deviations by n (population variance σ²); the n−1 sample variance is introduced in Lesson 34.5 on inference.