Range & Interquartile Range (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
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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
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)
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 |
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.
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
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