Understanding Quartiles, Median and Interquartile Range

In statistics, quartiles are three points that divide the data set into four equal groups. Each group represents the one-fourth of the data set.

First quartile (Q₁), also known as lower quartile, splits the lower 25% of data. It is the middle value of lower half.

Second quartile (Q₂) which is more commonly known as median splits the data in half (50%). Median divides the data into a lower half and an upper half.

Third quartile (Q₃), also known as upper quartile, splits lowest 75% (or highest 25%) of data. It is the middle value of the upper half.

The first quartile is also known as 25^th percentile, the second quartile as 50^th percentile, and the third quartile as 75^th percentile.

(A percentile is the value of a variable below which a certain percent of observations fall)

The Interquartile range is from Q₁ to Q₃. It is the difference between lower quartile and upper quartile.

IQR = Q₃ – Q₁

First arrange the data in ascending order. Then find the median. . If there is an odd number of values in a data set, the median is the middle value [(n+1)^th term]. If there is an even number of values in a data set, the median is the average of the two middle values [(n/2)^th and (n/2 + 1)^th terms]. The median divides the list of data into lower half and upper half. The lower quartile of the first half and upper quartile of second half can be determined using the same steps as median depending on the number of values in the halves.

Example 1:

4, 7, 2, 9, 5, 4, 2, 6

First arrange the data in ascending order

2, 2, 4, 4, 5, 6, 7, 9

N = 8 (even)

Median = (4 + 5)/2 = 9/2 = 4.5

Lower half: 2, 2, 4, 4

Lower quartile (Q₁) = (2 + 4)/2 = 3

Upper half: 5, 6, 7, 9

Upper quartile (Q₃) = (6 + 7)/2 = 6.5

Interquartile range = 6.5 – 3 = 3.5

Example 2:

12, 6, 7, 13, 11, 10, 4, 10, 12

N = 9 (odd)

In ascending order

4, 6, 7, 10, 10, 11, 12, 12, 13

Median = 10

Lower half: 4, 6, 7, 10

Lower quartile: (6 + 7)/2 = 6.5

Upper half: 11, 12, 12, 13

Upper quartile: (12 + 12)/2 = 12

Interquartile range = 12 – 6.5 = 5.5

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