01 · A negative minimum
Find the range of −8, −3, 1, 5, 7.
Hint
Subtract −8 from 7.
Worked solution
7 − (−8) = 15.
Understand · explore · practise
Range and interpercentile spread
Compare range, interquartile range and interpercentile ranges, including extreme values, grouped estimates and cautious contextual interpretation.
Before you startQuartiles, percentiles and grouped interpolation.
01 / Separate centre from spread
Two groups can have the same centre and different variation.
Range = maximum − minimum. IQR = Q₃ − Q₁.
Both are differences in the original variable’s units.
The range uses the two extremes. The IQR measures the distance between the lower and upper quartiles. Always state which measure you are comparing.
How far does one extreme reach?Explore
Values: 2, 4, 6, 8, 10, 12, 14, 20.
Range = 20 − 2 = 18.
Q₁ = (4 + 6)/2 = 5; Q₃ = (12 + 14)/2 = 13.
IQR = 13 − 5 = 8. Median = 9.
Here the largest value changes without changing the ranks used for the quartiles. That explains this example’s unchanged IQR; it is not a universal guarantee for every list.
02 / Use the observed extremes
Range is a distance, not an interval.
Temperatures: −4, −1, 2, 3, 6 °C.Worked example
Minimum = −4; maximum = 6
Negative values still have an ordinary ordering.
Range = 6 − (−4) = 10 °C
Subtract the minimum, including its sign.
The observed interval is −4 to 6 °C
That interval is not the numerical range.
02 · Frequency zero
Values 0, 2, 4, 9 have frequencies 0, 3, 2, 1. Find the observed range.
Hint
Zero does not occur.
Worked solution
The minimum observation is 2, not 0; maximum 9. Range = 7.
03 / Subtract quartile values
Do not subtract their rank positions.
Ordered values: 2, 4, 6, 8, 10, 12, 14, 20.Worked example
Q₁ = (4 + 6)/2 = 5; Q₃ = (12 + 14)/2 = 13
Use this course’s stated listed-data convention.
IQR = 13 − 5 = 8
Compare values, not ranks.
Median = (8 + 10)/2 = 9
The centre is a separate summary.
03 · Supplied quartiles
Journey times have Q₁ = 18 and Q₃ = 31 minutes. Find and interpret the IQR.
Hint
Take the difference.
Worked solution
IQR = 13 minutes: the interval between the two quartile values is 13 minutes wide.
04 · Calculate from a list
For 1, 3, 4, 6, 7, 8, 10, 12, 14, 19, use this course’s rule to find IQR.
Hint
Q₁ is rank 3; Q₃ is rank 8.
Worked solution
Q₁ = 4, Q₃ = 12; IQR = 8.
04 / Why IQR is often more resistant
Explain the mechanism rather than memorising a slogan.
Watch: a changing maximum and fixed quartiles
Pause, replay or seek freely. The notes explain the same idea and stay in view.
In the model, raising the maximum changes the range directly, while the quartile positions continue to use unchanged values. The IQR is often more resistant to extremes than the range. However, quartiles can change if edited values enter their positions or if the list is small.
05 · Model comparison
Raise the model maximum from 20 to 100. State the old and new range and IQR.
Hint
The minimum stays 2; quartile values stay 5 and 13.
Worked solution
Range rises from 18 to 98. IQR remains 8 in this particular list.
06 · A small-list counterexample
Under this lesson’s quartile rule, compare the IQRs of 0, 1, 2, 3 and 0, 1, 2, 100.
Hint
For four values, average ranks 1–2 for Q₁ and 3–4 for Q₃.
Worked solution
First: Q₁ = 0.5, Q₃ = 2.5, IQR = 2. Second: Q₁ = 0.5, Q₃ = 51, IQR = 50.5. Thus “IQR never changes when an extreme changes” is false.
05 / Choose a wider central interval
Name both percentiles.
The interdecile range is P₉₀ − P₁₀. More generally, an interpercentile range is the upper percentile value minus the lower percentile value. The 10th-to-90th interval summarises the central 80% in a continuous model, but tied finite data need not place exactly 80% strictly inside the endpoints.
07 · Interdecile range
A summary gives P₁₀ = 8 and P₉₀ = 35 seconds. Calculate the interdecile range.
Hint
Subtract the tenth percentile value.
Worked solution
35 − 8 = 27 seconds.
08 · Another central interval
P₅ = 2.5 and P₉₅ = 17.2 kg. Give the interpercentile range and the central percentage it targets.
Hint
Use the percentile labels for percentage; values for width.
Worked solution
Width = 17.2 − 2.5 = 14.7 kg. The interval targets the central 95 − 5 = 90% in a continuous distribution.
06 / Retain the uncertainty of grouping
Estimated quartiles give an estimated IQR.
For grouped data, interpolate each quartile using its own class and then subtract. The exact sample range is usually unknown: class boundaries are not necessarily observed extremes. A boundary span can give a useful upper bound or a rough range estimate, but must be labelled.
09 · Grouped IQR
Interpolated quartiles are 11.25 and 27.5 minutes. Find the estimated IQR.
Hint
Subtract the estimates.
Worked solution
27.5 − 11.25 = 16.25 minutes. It remains an estimate based on the within-class interpolation assumption.
10 · Range from occupied classes
The lowest occupied interval is [10,20) and the highest is [40,50). Is the observed range exactly 40?
Hint
The minimum is at least 10 and the maximum is below 50.
Worked solution
No. The boundary span is 50 − 10 = 40, but the actual range is less than 40. Since a lowest-class observation is below 20 and a highest-class observation is at least 40, the actual range is greater than 20. Thus 20 < range < 40; the exact value is unknown.
07 / Compare like with like
A smaller spread does not automatically mean better performance.
11 · Compare two services
Service A has median journey time 22 minutes and IQR 6; B has median 19 and IQR 11. Compare the summaries.
Hint
Discuss centre and spread separately.
Worked solution
B has the lower median journey time, while A has the smaller spread between quartiles. These summaries do not show every journey or establish that one service causes a particular outcome.
12 · Units matter
Dataset A has IQR 0.2 hours and B has IQR 10 minutes. Which IQR is smaller?
Hint
Convert to a common unit.
Worked solution
A: 0.2×60 = 12 minutes. B: 10 minutes, so B has the smaller IQR. Comparing 0.2 with 10 without units would mislead.
08 / State the measure and its limits
Centre, variation and context belong together.
Report the numerical difference with units, say whether it is exact or estimated, and explain what it describes. Check quartile conventions and measurement units before comparing groups.
Section 1 of 8 · Separate centre from spread