Statistics · Measures of Location and Spread · Year 1
Measures of Spread
Two datasets can share exactly the same mean and still look completely different — a measure of spread is what captures that difference.
9 min read4 worked examples
Range and interquartile range
A measure of spread (also called a measure of dispersion or variation) describes how spread out a dataset is. The simplest is the range, the difference between the largest and smallest values. The interquartile range (IQR) is the difference between the upper and lower quartiles:
$\text{IQR} = Q_3 - Q_1.$
Why the IQR is often preferred to the range
The range depends only on the two most extreme values, so a single unusually high or low reading distorts it completely. The IQR ignores the top and bottom quarter of the data, making it far more resistant to extreme values — a property called robustness. Its cost is that it only describes the spread of the middle 50% of the data, saying nothing about the tails.
Interpercentile range
More generally, an interpercentile range is the difference between two chosen percentiles. The 10th to 90th interpercentile range is commonly used, since it's not affected by extreme values but still reflects the spread of 80% of the data — a useful middle ground between the range (100% of the data, but not robust) and the IQR (only 50% of the data, very robust).
Estimating spread for grouped data
For grouped or continuous data, the same quantities are estimated using the interpolated quartile and percentile positions from the previous lesson — the range, IQR or interpercentile range is then just the difference between the two interpolated estimates.
Worked example 1
Range and IQR from summary values
A dataset of 17 ordered values has minimum $5$, $Q_1=11$, median $17$, $Q_3=24$, and maximum $45$. Find the range and the interquartile range.
MethodSubtract the relevant pair of values
1
Range $= 45-5=40$.
Maximum minus minimum.
2
$\text{IQR} = Q_3-Q_1 = 24-11=13$.
Upper quartile minus lower quartile.
▪ solved
Look deeper
The range ($40$) is more than three times the IQR ($13$), showing the extreme values sit much further from the bulk of the data than the middle 50% of values are from each other — a sign that at least one value may be an outlier.
Worked example 2
Interpercentile range from given percentiles
The 10th and 90th percentiles of the finishing times of 200 marathon runners are $P_{10}=3\text{h }42\text{min}$ and $P_{90}=5\text{h }21\text{min}$. Find the 10–90 interpercentile range in minutes.
10–90 interpercentile range $=321-222=99$ minutes.
Upper percentile minus lower percentile.
▪ solved
Worked example 3
IQR estimate from grouped continuous data
The masses, $m$ kg, of 80 parcels are grouped below.
Mass, $m$ (kg)
$0\le m<2$
$2\le m<4$
$4\le m<6$
$6\le m<10$
Frequency
10
28
25
17
Find an estimate for the interquartile range.
MethodInterpolate Q₁ and Q₃, then subtract
1
Cumulative frequencies: $10,\ 38,\ 63,\ 80$.
Running total after each class.
2
$Q_1$ position $=\dfrac{80}{4}=20$, in the class $2\le m<4$ (cumulative 10 before, 38 after).
20 lies between 10 and 38.
3
$Q_1 \approx 2+\dfrac{20-10}{38-10}\times 2 = 2+\dfrac{10}{28}\times 2 \approx 2.71$ kg.
Interpolation formula.
4
$Q_3$ position $=\dfrac{3(80)}{4}=60$, in the class $4\le m<6$ (cumulative 38 before, 63 after).
60 lies between 38 and 63.
5
$Q_3 \approx 4+\dfrac{60-38}{63-38}\times 2 = 4+\dfrac{22}{25}\times 2 \approx 5.76$ kg.
Same interpolation method, different class.
6
$\text{IQR} \approx 5.76-2.71 = 3.05$ kg.
Q₃ minus Q₁.
▪ solved
Worked example 4
10–90 interpercentile range from grouped data · Challenge
Using the same parcel mass data as Worked Example 3, find an estimate for the 10th to 90th interpercentile range, and explain one advantage this has over using the range.
MethodInterpolate P₁₀ and P₉₀, then subtract and compare
1
$P_{10}$ position $=0.10\times 80=8$, in the class $0\le m<2$ (cumulative 0 before, 10 after).
8 lies between 0 and 10.
2
$P_{10} \approx 0+\dfrac{8-0}{10-0}\times 2 = 1.6$ kg.
Interpolation formula.
3
$P_{90}$ position $=0.90\times 80=72$, in the class $6\le m<10$ (cumulative 63 before, 80 after).
72 lies between 63 and 80.
4
$P_{90}\approx 6+\dfrac{72-63}{80-63}\times 4 = 6+\dfrac{9}{17}\times 4 \approx 8.12$ kg.
Same method, different class and width.
5
10–90 interpercentile range $\approx 8.12-1.6=6.52$ kg.
P₉₀ minus P₁₀.
6
Unlike the range (which would use the extreme class boundaries $0$ and $10$), this measure excludes the most extreme 10% at each end, so it isn't distorted by one or two unusually light or heavy parcels.
Advantage: robustness while still using most of the data.
▪ solved
Look deeper
The 10–90 interpercentile range ($6.52$ kg) sits between the IQR ($3.05$ kg) and the full range ($10-0=10$ kg) — exactly as expected, since it's based on a wider slice of the data (80%) than the IQR (50%) but still excludes the most extreme values that inflate the full range.
Common mistake
Don't confuse the interquartile range with the "interpercentile range" as if they're unrelated — the IQR is a specific interpercentile range, the 25th to 75th. Always check which two percentiles (or quartiles) a question is asking you to use before subtracting.
Check yourself
A dataset has $Q_1=18$, $Q_3=31$, minimum $6$ and maximum $52$. Find the range and the interquartile range, and state which is more affected by extreme values. (Hint: subtract the relevant pair for each measure.)
Method
1
Range $=52-6=46$.
2
$\text{IQR}=31-18=13$.
Q₃ minus Q₁.
3
The range is more affected by extreme values, since it uses the minimum and maximum directly.