Range and Measures of Spread

Averages tell you about the centre of a data set, but you also need to describe how spread out the data is. Edexcel frequently asks you to compare two distributions using an average and a measure of spread.

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Key Vocabulary

Term	Definition
Range	Largest value − smallest value
Interquartile range (IQR)	Upper quartile ($Q_3$Q3​) − Lower quartile ($Q_1$Q1​); the range of the middle 50% of data
Lower quartile ($Q_1$Q1​)	The value one-quarter of the way through the ordered data
Upper quartile ($Q_3$Q3​)	The value three-quarters of the way through the ordered data
Outlier	A value that is unusually large or small compared to the rest

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The Range

Range = largest value − smallest value

Worked Example 1

Find the range of: 3, 5, 7, 8, 12, 15, 42.

Solution: Range = 42 − 3 = 39.

The range is simple but has a major weakness: it is affected by outliers (extreme values). Here, 42 makes the range large even though most values are clustered between 3 and 15.

Worked Example 2

Daily maximum temperatures (°C) for one week were: 18, 21, 19, 22, 20, 17, 23. Find the range.

Solution: Largest = 23, smallest = 17. Range = 23 − 17 = 6 °C.

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Quartiles and Interquartile Range from Listed Data

For n ordered values:

• $Q_1$Q1​ position = (n + 1) ÷ 4
• $Q_2$Q2​ (median) position = (n + 1) ÷ 2
• $Q_3$Q3​ position = 3(n + 1) ÷ 4

Worked Example 3

Find $Q_1$Q1​, $Q_2$Q2​ (median), $Q_3$Q3​ and the IQR for: 2, 4, 5, 7, 8, 10, 11, 13, 15, 18, 20.

Solution:

• n = 11
• $Q_1$Q1​ position = (11+1) ÷ 4 = 3rd value → $Q_1$Q1​ = 5
• $Q_2$Q2​ position = (11+1) ÷ 2 = 6th value → $Q_2$Q2​ = 10
• $Q_3$Q3​ position = 3(11+1) ÷ 4 = 9th value → $Q_3$Q3​ = 15
• IQR = $Q_3 - Q_1$Q3​−Q1​ = 15 − 5 = 10

Worked Example 4

Find the IQR of: 12, 15, 17, 20, 22, 24, 28, 30, 35.

Solution:

• n = 9; data already ordered.
• $Q_1$Q1​ position = (9+1) ÷ 4 = 2.5 → $Q_1$Q1​ is halfway between 2nd (15) and 3rd (17) values = 16
• $Q_3$Q3​ position = 3(9+1) ÷ 4 = 7.5 → $Q_3$Q3​ is halfway between 7th (28) and 8th (30) values = 29
• IQR = 29 − 16 = 13

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Quartiles from a Frequency Table

Use cumulative frequencies to find the positions of $Q_1$Q1​, $Q_2$Q2​, $Q_3$Q3​.

Worked Example 5

Test scores for 40 students:

Score	1	2	3	4	5
Frequency	5	8	12	10	5
Cumulative freq.	5	13	25	35	40

Find $Q_1$Q1​, $Q_2$Q2​, $Q_3$Q3​ and the IQR.

Solution:

• $Q_1$Q1​ position = (40 + 1) ÷ 4 = 10.25th value → lies in score 2 row (CF reaches 13) → $Q_1$Q1​ = 2
• $Q_2$Q2​ position = (40 + 1) ÷ 2 = 20.5th value → lies in score 3 row (CF reaches 25) → $Q_2$Q2​ = 3
• $Q_3$Q3​ position = 3(40 + 1) ÷ 4 = 30.75th value → lies in score 4 row (CF reaches 35) → $Q_3$Q3​ = 4
• IQR = 4 − 2 = 2

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Identifying Outliers

A common rule (used on Edexcel papers) is:

• An outlier is any value less than $Q_1$Q1​ − 1.5 × IQR or greater than $Q_3$Q3​ + 1.5 × IQR.

Worked Example 6

For a data set $Q_1$Q1​ = 5, $Q_3$Q3​ = 15, IQR = 10. Which of the following are outliers: −12, 8, 25, 32?

Solution:

• Lower fence: $Q_1$Q1​ − 1.5 × IQR = 5 − 15 = −10
• Upper fence: $Q_3$Q3​ + 1.5 × IQR = 15 + 15 = 30
• Anything below −10 or above 30 is an outlier.
• −12 < −10 → outlier
• 8 is between −10 and 30 → not an outlier
• 25 is between −10 and 30 → not an outlier
• 32 > 30 → outlier

Worked Example 7

A data set has $Q_1$Q1​ = 16, $Q_3$Q3​ = 29, IQR = 13 (from Worked Example 4). Is 50 an outlier?

Solution: Upper fence = 29 + 1.5 × 13 = 29 + 19.5 = 48.5. Since 50 > 48.5, yes, 50 is an outlier.

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Comparing Distributions

When comparing two data sets, Edexcel expects you to:

1. Compare an average (mean or median) — say which is higher/lower and what this tells you in context.
2. Compare a measure of spread (range or IQR) — say which is larger/smaller and what this tells you about consistency.

Worked Example 8

Two football teams recorded the following goals-per-match statistics over a season:

	Team A	Team B
Median goals per match	2	3
IQR	1	4

Compare the two teams.

Solution:

• Team B has a higher median (3 > 2), so on average Team B scores more goals per match.
• Team A has a smaller IQR (1 < 4), so Team A's goal-scoring is more consistent — their results are closer to the median. Team B's scoring varies more from match to match.

Edexcel exam tip: You MUST relate your comparison back to the context. Don't just say "Team A has a smaller IQR" — say what it means about the goals/marks/times etc.

Worked Example 9

Two shops record daily customer numbers:

	Shop X	Shop Y
Median	120	95
IQR	15	40

Compare the two shops.

Solution:

• Shop X has a higher median (120 > 95), so on average Shop X gets more customers per day.
• Shop X also has a smaller IQR (15 < 40), so Shop X's daily customer numbers are more consistent — Shop Y is much more variable.

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Measures of Spread from Grouped Data

For grouped data you cannot find exact quartiles from the table alone — you use a cumulative frequency graph (see Lesson 7). However, you can:

• Identify the class containing $Q_1$Q1​ and the class containing $Q_3$Q3​ using cumulative frequencies.
• Estimate IQR from the graph by reading off $Q_1$Q1​ and $Q_3$Q3​ values.

For grouped data on a CF curve you typically use $Q_1$Q1​ at n/4, median at n/2, $Q_3$Q3​ at 3n/4 (without the +1).

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Worked Exam-Style Example

Worked Example 10

Two classes took the same maths test. Their results are summarised below.

	Class X	Class Y
Mean	62	58
Range	45	22

(a) Compare the performance of the two classes. (b) The teacher says "Class X did better." Do you agree? Explain your answer.

Solution:

(a) Class X has a higher mean (62 > 58), so on average Class X scored higher marks. Class Y has a smaller range (22 < 45), so Class Y's marks were more consistent and less spread out.