6.2 - Spread, quartiles and interquartile range

6.2 - Spread, quartiles and interquartile range

An average describes a centre but does not tell you how much observations vary. Calculate quartiles and interquartile range from discrete data, then compare spread and explain the effect of an extreme value.

What spread measures (Higher)

Can changing the largest value from 1717 to 5757 leave a measure of spread unchanged? It depends on which part of the data that measure describes.

You already know

Order observations to find a median. The range is largest minus smallest; it has the same unit as the data. A range of zero means every observation is equal.

A median locates the middle, but it does not tell you whether observations cluster close to it. Spread adds this missing information.

Measure of spread

A number describing how much observations vary. Range describes the full span; interquartile range describes the span of the central part of the ordered data.

For repeated measurements, a smaller spread can indicate more consistency by that measure. It does not automatically mean a lower average or a better outcome.

Worked example 1: equal centres, different spreads

Question. Compare the medians and ranges of A: 8,9,10,11,128,9,10,11,12 and B: 0,5,10,15,200,5,10,15,20.

Plan. Locate each middle value, then subtract each pair of endpoints.

median A=10third of five valuesmedian B=10third of five valuesrange A=12−8=4A’s full spanrange B=20−0=20B’s full span\begin{aligned} \text{median A}&=10 &&\text{third of five values}\\ \text{median B}&=10 &&\text{third of five values}\\ \text{range A}&=12-8=4 &&\text{A's full span}\\ \text{range B}&=20-0=20 &&\text{B's full span} \end{aligned}

Why calculate both centre and spread? The equal medians hide the much wider spacing in B.

Answer. Both medians are 1010; A has less spread by range, 44 compared with 2020.

[DIAGRAM: asset_slug: m091_spread; caption: Two five-value datasets have the same median but different ranges; description: A has values eight to twelve close to median ten, while B has zero five ten fifteen twenty spread across a larger interval; file: m091_spread.png; recommended_method: matplotlib]
Two five-value datasets have the same median but different ranges

Check. Both lists have two positions on either side of 1010, but their endpoints differ.

Change only B's largest value from 2020 to 4040. The new value remains largest, so the order of the other observations stays fixed.

Worked example 2: move one endpoint

Question. Find B's revised median and range, then compare it with unchanged A.

Plan. Keep the middle position and update the largest endpoint.

revised B:0,5,10,15,40only maximum changedmedian B=10same third valuerange B=40−0=40new endpoint difference\begin{aligned} \text{revised B}&:0,5,10,15,40 &&\text{only maximum changed}\\ \text{median B}&=10 &&\text{same third value}\\ \text{range B}&=40-0=40 &&\text{new endpoint difference} \end{aligned}

Why is the median unchanged? The observation in the middle position did not change.

Answer. B still has median 1010, but its range is 4040; A still has the smaller range, 44.

Check. B's maximum rose by 2020, and its range also rose by 2020.

Compare the same variable in the same units. Convert minutes and seconds before comparing numerical spreads. A spread describes these observations, rather than guaranteeing the size of future variation.

Watch out

The conclusion “equal medians mean equal spread” is false: the first example has ranges 44 and 2020. A median supplies a centre, not a spread.

The check changes the endpoints again while keeping the middle values equal. Use the range relationship; it is not in the Higher formulae sheet.

Quartiles and interquartile range (Higher)

Quartiles locate the lower and upper edges of the central portion of ordered data.

The lower quartile, Q1Q_1, lies near one quarter of the way through the ordered observations. The upper quartile, Q3Q_3, lies near three quarters. Their difference measures central spread, rather than average.

Interquartile range

IQR=Q3−Q1\mathrm{IQR}=Q_3-Q_1

The IQR has the same unit as the data. Not in the Higher formulae sheet: learn it.

Small discrete lists have several quartile conventions. Here we use medians of halves: order every observation; split an even-sized list equally; for an odd-sized list exclude the overall median, then split the rest equally. Find the median of each half, averaging its two middle values if needed. Our tasks state that convention. Follow a different stated convention when a question gives one; calculator quartiles can use another rule. Do not mix rules in one comparison.

Worked example 3: quartiles for eight observations

Question. Find Q1Q_1, Q3Q_3 and the IQR of 1,4,5,8,10,12,15,171,4,5,8,10,12,15,17, using medians of halves.

Plan. Split the ordered list into equal halves, then find their medians.

overall median=8+102=9middle pair of whole listlower half:1,4,5,8first four observationsupper half:10,12,15,17last four observationsQ1=4+52=4.5median of lower halfQ3=12+152=13.5median of upper halfIQR=13.5−4.5=9upper minus lower\begin{aligned} \text{overall median}&=\frac{8+10}{2}=9 &&\text{middle pair of whole list}\\ \text{lower half}&:1,4,5,8 &&\text{first four observations}\\ \text{upper half}&:10,12,15,17 &&\text{last four observations}\\ Q_1&=\frac{4+5}{2}=4.5 &&\text{median of lower half}\\ Q_3&=\frac{12+15}{2}=13.5 &&\text{median of upper half}\\ \mathrm{IQR}&=13.5-4.5=9 &&\text{upper minus lower} \end{aligned}

Why average pairs within each half? Each half has four values, so each has two middle positions.

Answer. Q1=4.5Q_1=4.5, Q3=13.5Q_3=13.5, IQR 99.

Check. 4.5≤9≤13.54.5\leq9\leq13.5; the IQR 99 is between zero and the range 17−1=1617-1=16.

Change only the observation count by inserting one value, 99, at the centre. The odd-list convention now matters.

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