GCSEMathsWJECMaths & Numeracy (Double Award)HigherRevision NotesStatistics & ProbabilityAverages Ranges & DataRange & Interquartile Range

Range & Interquartile Range (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

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Range & IQR

What is the range?

  • The range is the difference between the highest value and the lowest value

    • range = highest - lowest

      • For example, the range of 1, 2, 5, 8 is 8 - 1 = 7

  • It measures how spread out the data is

    • Ranges of different data sets can be compared to see which is more spread out

    • The range of a data set can be affected by very large or small values

  • Be careful with negatives

    • The range of -2, -1, 0, 4 is 4 - (-2) = 6

How do I know when to use the range?

  • The range is a simple measure of how spread out the data is

    • The range does not measure an average value

  • It should not be used if there are any extreme values (outliers)

    • For example, the range of 1, 2, 5, 80 is 80 - 1 = 79

      • This is not a good measure of spread

      • The range is affected by extreme values

What are quartiles?

  • The median splits the data set into two parts

    • Half the data is less than the median

    • Half the data is greater than the median

  • Quartiles split the data set into four parts

    • The lower quartile (LQ) lies a quarter of the way along the data (when in order)

      • One quarter (25%) of the data is less than the LQ

      • Three quarters (75%) of the data is greater than the LQ

    • The upper quartile (UQ) lies three quarters of the way along the data (when in order)

      • Three quarters (75%) of the data is less than the UQ

      • One quarter (25%) of the data is greater than the UQ

    • You may come across the median being referred to as the second quartile

How do I find the quartiles?

  • Make sure the data is written in numerical order

  • Use the median to divide the data set into lower and upper halves

    • If there are an even number of data values, then

      • the first half of those values are the lower half,

      • and the second half are the upper half

      • All of the data values are included in one or other of the two halves

    • If there are an odd number of data values, then

      • all the values below the median are the lower half

      • and all the values above the median are the upper half

      • The median itself is not included as a part of either half

  • The lower quartile is the median of the lower half of the data set

    • and the upper quartile is the median of the upper half of the data set

  • Find the quartiles in the same way you would usually find the median

    • just restrict your attention to the relevant half of the data

How do I find the quartiles from grouped data?

  • If there are n data values in total

    • The lower quartile will be the open parentheses n over 4 close parentheses to the power of th data value

    • The upper quartile will be the open parentheses fraction numerator 3 n over denominator 4 end fraction close parentheses to the power of th data value

  • You may have to use linear interpolation to find the quartiles

    • It is the same method you use for finding the median from grouped data

      • You just find the open parentheses n over 4 close parentheses to the power of th or open parentheses fraction numerator 3 n over denominator 4 end fraction close parentheses to the power of th value instead of the open parentheses n over 2 close parentheses to the power of th value

    • The values for the quartiles will be estimates when you do this

What is the interquartile range (IQR)?

  • The interquartile range (IQR) is the difference between the upper quartile (UQ) and the lower quartile (LQ)

    • Interquartile range (IQR) = upper quartile (UQ) - lower quartile (LQ)

  • The IQR measures how spread out the middle 50% of the data is

    • The IQR is not affected by extreme values in the data

Examiner Tips and Tricks

If asked to find the range in an exam, make sure you show your subtraction clearly (don't just write down the answer). You may gain a method mark for this.

Worked Example

Find the range of the data in the table below.

3.4

4.2

2.8

3.6

9.2

3.1

2.9

3.4

3.2

3.5

3.7

3.6

3.2

3.1

2.9

4.1

3.6

3.8

3.4

3.2

4.0

3.7

3.6

2.8

3.9

3.1

3.0

Answer:

Range = highest value - lowest value

9.2 - 2.8

The range is 6.4

Worked Example

A naturalist studying crocodiles has recorded the numbers of eggs found in a random selection of 20 crocodile nests

31      32      35      35      36      37      39      40      42      45

46      48      49      50      51      51      53      54      57      60

Find the lower and upper quartiles for this data set.

Answer:

There are 20 data values (an even number)
So the lower half will be the first 10 values
The lower quartile is the median of that lower half of the data

31      32      35      35      36      37      39      40      42      45

So the lower quartile is midway between 36 and 37 (i.e. 36.5)

Do the same thing with the upper half of the data to find the upper quartile
The upper quartile is the median of the upper half of the data

46      48      49      50      51      51      53      54      57      60

So the upper quartile is midway between 51 and 51 (i.e. 51)

Lower quartile = 36.5
Upper quartile = 51

Worked Example

A student collected data about the length of time (x hours) students in his school spent listening to music in a given week. He collected data from 50 students in total.  The following table summarises the data:

Time spent, x (hours)

Number of students

0 ≤ x ≤ 10

3

10 < x ≤ 20

19

20 < x ≤ 30

12

30 < x ≤ 40

10

40 < x ≤ 50

5

50 < x ≤ 60

1

Work out an estimate for the interquartile range of the amount of time spent listening to music by the students.

Answer:

There are 50 students in total

The lower quartile will be the open parentheses 50 over 4 close parentheses to the power of th equals 12.5 to the power of th value

Add a cumulative frequency column to the table to identify which interval this lies in

Time spent, x (hours)

Number of students

Cumulative Frequency

0 ≤ x ≤ 10

3

3

10 < x ≤ 20

19

22

20 < x ≤ 30

12

34

30 < x ≤ 40

10

44

40 < x ≤ 50

5

49

50 < x ≤ 60

1

50

The lower quartile is in the 10 < x ≤ 20 interval

Use linear interpolation to find an estimate for the lower quartile

It will be 12.5 minus 3 equals 9.5 values (out of 19 values) into this interval

open parentheses fraction numerator 12.5 minus 3 over denominator 19 end fraction cross times 10 close parentheses plus 10 equals 15

Repeat for the upper quartile

The upper quartile will be the open parentheses fraction numerator 3 cross times 50 over denominator 4 end fraction close parentheses to the power of th equals 37.5 to the power of th value

Therefore it lies in the 30 < x ≤ 40 interval

It will be 37.5 minus 34 equals 3.5 values (out of 10 values) into this interval

open parentheses fraction numerator 37.5 minus 34 over denominator 10 end fraction cross times 10 close parentheses plus 30 equals 33.5

Find the interquartile range by finding the difference between the upper and lower quartiles

33.5 - 15 = 18.5

Remember to state the units

IQR = 18.5 hours

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.