Quartiles & Percentiles (Edexcel GCSE Statistics)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
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 3rd value and the UQ is the 9th value
for 12 data values, and
so the LQ is the 3.25th value and the UQ is the 9.75th value
for 13 data values, and
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
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 1412.5 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 3635.25 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 and
There are 20 data values, so n here is 20
so the LQ is the 5.25th value
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
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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