Quartiles & Percentiles (Edexcel GCSE Statistics)

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  value

    • the upper quartile is the  value

  • e.g. for 11 data values, and

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

  • for 12 data values, and

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

  • for 13 data values, and

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

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

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

    • then the LQ is

• 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 3.25^th value, and the 3^rd and 4^th 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 3^rd and 4^th values

    • so 12.5 is the LQ

  • Or if the UQ is the 9.75^th value, and the 9^th and 10^th 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 9^th and 10^th 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

Percentiles & Deciles

What are percentiles?

• Percentiles divide a data set into 100 equal parts

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

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

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

  • Note that

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

    • the 50^th percentile is the same as the median

    • the 75^th 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 2.5^th 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 99^th percentile)

    • with the median income (the 50^th percentile)

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

What are deciles?

• Deciles divide a data set into 10 equal parts

  • It's easiest to think of deciles in terms of percentiles

    • The 1st decile is the same as the 10^th percentile

    • The 2nd decile is the same as the 20^th percentile

    • etc.

    • (The 5^th decile is also the same as the median)

• Deciles can be calculated in the same way as the corresponding percentiles