Quartiles & Percentiles (Edexcel GCSE Statistics)

Revision Note

Quartiles

What are quartiles?

  • The median splits the data set into two parts

    • Half the data is less than the median (and half the data is greater than it)

  • 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 (and three quarters is greater than it)

    • 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 (and one quarter is greater than it)

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

How do I find the quartiles?

  • There are two different methods for finding the quartiles

    • Use whichever method you find easiest

  • In some cases the two methods will give slightly different answers

    • Either answer will receive full marks on the exam

    • Be sure to show your working so the examiner knows what you have done

Method 1

  • 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 the 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 find the median for any other data set

    • just restrict your attention to the lower or upper half of the data accordingly

Method 2

  • The quartiles can also be given in formula form:

    • For n data values

      • the lower quartile is the open parentheses fraction numerator n plus 1 over denominator 4 end fraction close parentheses to the power of th value

      • the upper quartile is the open parentheses 3 cross times fraction numerator n plus 1 over denominator 4 end fraction close parentheses to the power of th value

    • e.g. for 11 data values, fraction numerator 11 plus 1 over denominator 4 end fraction equals 3 and 3 cross times fraction numerator 11 plus 1 over denominator 4 end fraction equals 9

      • so the LQ is the 3rd value and the UQ is the 9th value

    • for 12 data values, fraction numerator 12 plus 1 over denominator 4 end fraction equals 3.25 and 3 cross times fraction numerator 12 plus 1 over denominator 4 end fraction equals 9.75

      • so the LQ is the 3.25th value and the UQ is the 9.75th value

    • for 13 data values, fraction numerator 13 plus 1 over denominator 4 end fraction equals 3.5 and 3 cross times fraction numerator 13 plus 1 over denominator 4 end fraction equals 10.5

      • so the LQ is the 3.5th value and the UQ is the 10.5th value

  • For a '.5' value, find the midpoint between the values on either side

    • e.g. if the LQ is the 3.5th value, and the 3rd and 4th values are 13 and 16

      • then the LQ is fraction numerator 13 plus 16 over denominator 2 end fraction equals 14.5

  • For a '0.25' or '0.75' value you need to find the value a quarter or three-quarter ways between the values on either side

    • e.g. if the LQ is the 3.25th value, and the 3rd and 4th values are 12 and 14

      • then divide the interval into 4 equal parts:
        12      12.5      13      13.5      14

      • 12.5 is a quarter of the way between the 3rd and 4th values

      • so 12.5 is the LQ

    • Or if the UQ is the 9.75th value, and the 9th and 10th values are 33 and 36

      • then divide the interval into 4 equal parts
        33      33.75      34.5      35.25      36

      • 35.25 is three quarters of the way between the 9th and 10th values

      • so 35.25 is the UQ

What about quartiles for grouped data?

  • Quartiles for grouped data will normally be calculated using a cumulative frequency diagram

    • See the 'Interpreting Cumulative Frequency Diagrams' note in the 'Cumulative Frequency Charts' revision note for how this works

  • Quartiles for grouped data can also be calculated using the method of linear interpolation

    • This is similar to finding the median for grouped data

    • See the 'Linear Interpolation' revision note for how this method works

Examiner Tips and Tricks

  • You can use either method to find the quartiles

    • But be sure to show your working to get full marks

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.

Method 1

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

Method 2

Use the open parentheses fraction numerator n plus 1 over denominator 4 end fraction close parentheses to the power of th and open parentheses 3 cross times fraction numerator n plus 1 over denominator 4 end fraction close parentheses to the power of th
There are 20 data values, so n here is 20

fraction numerator 20 plus 1 over denominator 4 end fraction equals 5.25 so the LQ is the 5.25th value

3 cross times fraction numerator 20 plus 1 over denominator 4 end fraction equals 3 cross times 5.25 equals 15.75 so the UQ is the 15.75th value

The 5th value is 36 and the 6th value is 37
Divide that interval into 4 equal parts

36      36.25      36.5      36.75      37

36.25 is one quarter of the way from 36 to 37, so that is the LQ

The 15th and 16th values are both 51, so that means the UQ is also 51

Lower quartile = 36.25
Upper quartile = 51

Note that Method 2 gives a different answer for the LQ
Both answers are 'correct', and both answers would get full marks on the exam

Percentiles

What are percentiles?

  • Percentiles divide a data set into 100 equal parts

    • n% of the data values will be less than the nth percentile

      • e.g. 10% of data values will be less than the 10th percentile (and 90% will be greater than it)

      • 99% of data values will be less than the 99th percentile (and 1% will be greater than it)

    • Note that

      • the 25th percentile is the same as the lower quartile

      • the 50th percentile is the same as the median

      • the 75th percentile is the same as the upper quartile

    • Also note that percentiles don't need to be whole numbers

      • e.g. we can talk about the 2.5th percentile

      • 2.5% of the data will be less than that (and 97.5% will be greater than it)

  • Percentiles can be useful for discussing the distribution of data in a data set

    • e.g. in comparing incomes in the UK

      • you could compare the highest 1% of earners (the ones above the 99th percentile)

      • with the median income (the 50th percentile)

      • or the lowest 10% of earners (those below the 10th percentile)

  • Percentiles will normally be calculated using a cumulative frequency diagram

    • See the 'Interpreting Cumulative Frequency Diagrams' note in the 'Cumulative Frequency Charts' revision note for how this works

  • Percentiles can also be calculated using the method of linear interpolation

    • This is similar to finding the median for grouped data

    • See the 'Linear Interpolation' revision note for how this method works

Worked Example

The table shows information about the times, in minutes, taken by 48 students to complete a maths quiz.

Time (t minutes)

Frequency

8 < t ≤ 10

6

10 < t ≤ 12

24

12 < t ≤ 14

11

14 < t ≤ 16

6

16 < t ≤ 18

1

Find the class interval that contains the 65th percentile.

65% of data values will be below the 65th percentile

First we need to find what 65% of 48 (the total number of data values) is

48 cross times 65 over 100 equals 31.2

So the 65th percentile is the 31.2th data value
That means it lies between the 31st and 32nd data values

The first two class intervals together contain 6+24=30 data values
That means that the 31st and 32nd data values are both in the third class interval

The 65th percentile is in the 12 < t ≤ 14 class interval

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Roger B

Author: Roger B

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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.

Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.