Mean deviation and Standard deviation are two measures of spread that are based on the Arithmetic mean. These are more suitable for reasonably symmetric distributions. But for a not so symmetric distribution, these two mean-based measures are not suitable. A more appropriate measure is the Interquartile range (IQR).

When the data is arranged in ascending order,

i. the lower (or first) quartile (Q₁) is value of the [(1/4)(n + 1)]^th term in the series, where 'n' is the number of observations.

ii. the upper (or third) quartile (Q₃) is value of the [(3/4)(n + 1)]^th term in the series

iii. IQR is given by (Q₃ – Q₁)

iv. Semi-IQR is given by (Q₃ – Q₁)/2

It is also known as quartile deviation (QD)

The lower (or first) quartile (Q₁) is effectively the median of the lower half of the data. In terms of percentile, Q₁ is the 25th percentile.

The upper (or third) quartile (Q₃) is effectively the median of the upper half of the data. In terms of percentile, Q₃ is the 75th percentile.

The median itself can be considered as second quartile (Q₂). In terms of percentile, Q₂ is the 50th percentile.

Note: If 'n' is not an integer, the average of the nearest two integers is taken.

Consider the data set 5, 7, 9, 13, 5, 17, 6, 8, 11. Find the different quartile ranges.

The data set is 5, 7, 9, 13, 5, 17, 6, 8, 11

Arranging in ascending order

5, 5, 6, 7, 8, 9, 11, 13, 17

n = 9

{Since 'n' is odd, the median = [(n + 1)/2]th value.

i.e, 5th value in the series which is 8}

Lower quartile (Q₁) = [(1/4)(n + 1)]^th term

i.e, 2.5^th term

So we have to take mean of 2nd and 3rd terms.

∴ Q₁ = (5 + 6)/2 = 5.5

Upper quartile (Q₃) = [(3/4)(n + 1)]^th term

i.e, 7.5^th term

So we have to take mean of 7th and 8th terms.

∴ Q₃ = (11 + 13)/2 = 12

∴ IQR = (Q₃ – Q₁) = 12 – 5.5 = 6.5

Semi-IQR or Q.D. = (Q₃ – Q₁)/2 = 6.5/2 = 3.25