Inter-Quartile Range - Calculation of Inter-Quartile Range _IQR_ by hcj


									                  Calculation of Inter-Quartile Range (IQR)

Ultimately, the IQR must hold 50% of the data points, i.e.:

Calculation as follows:
The inter-quartile range is for use with the median and is more robust to outliers than
the range (which simply states the highest and lowest values). Therefore, by
providing information about the range in which 50% of the data lies we give
information regarding the overall spread of our data (i.e. having established that the
data is non-normal, we now want to illustrate whether there was only 1 outlier or if
the entire data set deviated from the mean).

       From the data set below the median = 5.5

       2, 3, 4, 5, 6, 7, 8, 9

The inter-quartile range is the range in which the middle 50% of the data points fall
(i.e. the distance between the 1st and 3rd quartile) and would be calculated as follows
for the above data set:

       25% of n         = 25% x (n+1)
                        = 0.25 x 9
                        = 2.25

        Therefore the 1st quartile lies between the 2nd and 3rd lowest observations (i.e.
3 &4). Because we know the distance (i.e. 2.25) we can calculate the precise location
of the 1st quartile as follows:

       1st quartile     = (0.25 x 2nd observation) + ((1- 0.25) x 3rd observation)
                        = (0.25 x 3) + (0.75 x 4)
                        = 3.75

Similarly, the third quartile can be derived through firstly calculating 75% of n:

       75% of n         = 75% x (n+1)
                        = 0.75 x 9
                        = 6.75                                 note, 6.75 means majority
                                                               (0.75) goes to higher value
…and then calculating the value which is the 6.75th lowest:

        3rd quartile    = ((1 - 0.75) x 6th observation) + ((0.75) x 7th observation)
                        = (0.25 x 7) + (0.75 x 8)
                        = 7.75
This IQR can now be stated to show the spread of scores about the median.
Importantly, it should not be given as a single number (i.e. 7.75-3.75=4) as this
suggests that the data points are equally distributed either side of the median. Instead
both numbers should be given, e.g: 5.5 (3.75 - 7.75)

(N.B: for and odd number of data points, exclude the median from calculation of quartiles)

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