Median & Interquartile Range (SQA National 5 Maths): Revision Note

Finding the median & interquartile range

What are the median and standard deviation of a data set?

• The median of a data set is a type of average

  • It is the middle value of a data set when the data values are written in numerical order

  • The mean is another type of average

• The interquartile range is a measure of spread

  • It tells you how spread out the middle half of a data set is

  • The standard deviation is another measure of spread

• The median and interquartile range are used together to analyse a set of data

  • The mean and standard deviation may also be used together to analyse a data set

  • But you should not use the mean with the interquartile range, or the median with the standard deviation

How do I find the median of a data set?

• The median is the middle value when you put values in size order

  • The median of 4, 2, 3 can be found by

    • ordering the numbers: 2, 3, 4

    • and choosing the middle value, 3

• If you have an even number of values, find the midpoint of the middle two values 

  • The median of 1, 2, 3, 4 is 2.5

    • 2.5 is the midpoint of 2 and 3

  • The midpoint is the sum of the two middle values divided by 2

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

  • The median is sometimes referred to as the second quartile

How do I find the quartiles of a data set?

• 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 interquartile range (IQR) of a data set?

• 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

  • Unlike the standard deviation, the IQR is not affected by extreme values in the data