Interquartile Range

Interquartile range is measure of dispersion which tells us about the “spread” of our data. It tells us the range within which 50% of our data lies. It is calculated by subtracting the lower quartile from the upper quartile.

If the value of the interquartile range is relatively small it means that there is very little variation in the central 50% of our data whereas if it is high that implies that there is more variation within the central values.

How to calculate Inter-quartile Range?

1. Arrange the data in ascending order.
2. Calculate the lower quartile using the formula Q₁=(N/4)^th term.
3. Calculate the upper quartile using the formula Q₃=(3N/4)^th term
4. Find the interquartile range by subtracting the lower quartile from the upper quartile, IQR=Q­₃-Q₁.

Example: Find the interquartile range for the following set of data:

11,13,19,23,28,32,37,39

Solution: Here we have 8 observations, that is, N=8

Lower quartile Q₁=(N/4)^th term= 2^nd term= 13

Upper quartile Q₃=(3N/4)^th term= 6^th term= 32

Interquartile range IQR= Q₃-Q₁= 32-13= 19

Coefficient of Quartile Deviation:

The Interquartile range is an absolute measure of dispersion. It is convenient to have a relative measure of dispersion which always lies between 0 and 1. In this case it is much easier to know if the spread of the data is high or not. If the value of coefficient of quartile deviation lies closer to 1 then the variation is high whereas if it lies closer to 0 then the variation is low.

The formula for coefficient of quartile deviation is:

Coefficient of quartile deviation= (Q­₃-Q₁)/( Q­₃+Q₁)

Continuing the above example we find,

Coefficient of quartile deviation= (Q­₃-Q₁)/( Q­₃+Q₁)= 32-13/32+13= 19/43 = 0.442 which is less than 0.5 and that tells us there is less variation in the given data.

Advantages of Interquartile Range:

There are other measures of dispersion as well such as standard deviation. But the advantage of interquartile range is that it is much easier to calculate as compared to the standard deviation.

We can also use interquartile range to find outliers which may be present in our data

The disadvantage is that interquartile range does not consider all values whereas the standard deviation considers all observations in the data.