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Lower quartile Median Upper quartile
( Q 1 ) ( Q 2 ) ( Q 3 )

Method: Calculating the quartiles

  1. Order the data from smallest to largest or from largest to smallest.
  2. Count how many data values there are in the data set.
  3. Divide the number of data values by 4. The result is the number of data values per group.
  4. Determine the data values corresponding to the first, second and third quartiles using the number of data values per quartile.

What are the quartiles of { 3 , 5 , 1 , 8 , 9 , 12 , 25 , 28 , 24 , 30 , 41 , 50 } ?

  1. { 1 , 3 , 5 , 8 , 9 , 12 , 24 , 25 , 28 , 30 , 41 , 50 }

  2. There are 12 values in the data set.

  3. 12 ÷ 4 = 3
  4. 1 3 5 8 9 12 24 25 28 30 41 50
    Q 1 Q 2 Q 3

    The first quartile occurs between data position 3 and 4 and is the average of data values 5 and 8. The second quartile occurs between positions 6 and 7 and is the average of data values 12 and 24. The third quartile occurs between positions 9 and 10 and is the average of data values 28 and 30.

  5. The first quartile = 6,5. ( Q 1 )

    The second quartile = 18. ( Q 2 )

    The third quartile = 29. ( Q 3 )

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Inter-quartile range

Inter-quartile Range

The inter quartile range is a measure which provides information about the spread of a data set, and is calculated by subtracting the first quartile from the third quartile, giving the range of the middle half of the data set, trimming off the lowest and highest quarters, i.e. Q 3 - Q 1 .

The semi-interquartile range is half the interquartile range, i.e. Q 3 - Q 1 2

A class of 12 students writes a test and the results are as follows: 20, 39, 40, 43, 43, 46, 53, 58, 63, 70, 75, 91. Find the range, quartiles and the Interquartile Range.

  1. 20 39 40 43 43 46 53 58 63 70 75 91
    Q 1 M Q 3
  2. The range = 91 - 20 = 71. This tells us that the marks are quite widely spread. (Remember, however, that 'wide' and 'large' are relative terms. If you are considering one hundred people, a range of 71 would be 'large', but if you are considering one million people, a range of 71 would likely be 'small', depending, of course, on what you were analyzing).

  3. i.e. M = 46 + 53 2 = 99 2 = 49 , 5

  4. i.e. Q 1 = 40 + 43 2 = 83 2 = 41 , 5

  5. i.e. Q 3 = 63 + 70 2 = 133 2 = 66 , 5

  6. The quartiles are 41,5, 49,5 and 66,5. These quartiles tell us that 25 % of the marks are less than 41,5; 50 % of the marks are less than 49,5 and 75 % of the marks are less than 66,5. They also tell us that 50 % of the marks lie between 41,5 and 66,5.

  7. The Interquartile Range = 66,5 - 41,5 = 25. This tells us that the width of the middle 50 % of the data values is 25.

  8. The Semi-interquartile Range = 25 2 = 12,5

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Percentiles

Percentiles

Percentiles are the 99 data values that divide a data set into 100 groups.

The calculation of percentiles is identical to the calculation of quartiles, except the aim is to divide the data values into 100 groups instead of the 4 groups required by quartiles.

Questions & Answers

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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