Range & Quartiles (Cambridge O Level Maths)

What are the range and interquartile range (IQR)?

• The three averages (mean, median and mode) measure what is called central tendency
  • they all give an indication of what is typical about the data
  • what lies roughly in the middle
• The range and interquartile range (IQR) measure how spread out the data is
  • These can only be applied to numerical data
  • Fortunately both are easy to work out!

How do I work out the range?

• The range is the difference between the highest data value and the lowest data value
  • It measures how spread out the data is
  • You can remember this as "Hi - Lo"

• There is one possible problem with the range
  • as it considers the highest and lowest values in the data set it could be influenced by anomalies (outliers) in the data
  • these may not be a true representation of how spread the rest of the data may be

How do I find the quartiles?

• The median splits the data set into two parts, lying half way along the data
  • As their name suggests, quartiles split the data set into four parts
    • The lower quartile (LQ) lies a quarter of the way along the data (when in order)
    • The upper quartile (UQ) lies three quarters of the way along the data
    • You may come across the median being referred to as the second quartile
• To find the quartiles first use the median to divide the data set into lower and upper halves
  
  • Make sure the data is put into numerical order first
  • 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
    • In this case, all the numbers in the data set 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
    • In this case, 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
• Sometimes you may also see the quartiles given in formula form
  
  • For n data values:
    • the lower quartile is the  value
    • the upper quartile is the  value
  • Using these can save finding the median and splitting the data into two halves

How do I work out the interquartile range (IQR)?

• This is the difference between the upper quartile (UQ) and the lower quartile (LQ)
  
  • So you need to find the quartiles first before you can calculate the interquartile range
• The interquartile range is  IQR = UQ - LQ
• The interquartile range considers the middle 50% of the data so is not affected by extreme values in the data
  • The range of a data set, on the other hand, can be affected by extremely large or small values