How to find Quartiles for Even and Odd Number of Data

The quartiles are three numbers that split our data into precisely four equal parts. They are the lower quartile (Q₁), the median (Q_2), and the upper quartile (Q₃).

The procedure to find the quartiles is different depending on whether the number of data values (denoted by N) is even or odd.

How to get Quartiles for Odd Number of Data:

1. Arrange the data in ascending order.
2. The middlemost term in the data set is the median (the second quartile Q₂). The middlemost term is the [(N+1)/2]^th term.
3. Split the data set into two equal halves at the median. Since N is odd, include the median in both the upper and lower halves of the data set.
4. The median of the lower data set gives us the lower quartile (Q₁).
5. The median of the upper half gives us the value of the upper quartile (Q₃).

Example 1:

Consider the following set of data values: 1, 2, 4, 5, 6, 7, 9, 11, 12.

Here the number of data values is equal to N = 7 (odd). The data is already in ascending order therefore the median is given by the [(7+1)/2]^th term = 4th term which is 6. So we conclude that,

Median = Second Quartile = Q₂ = 6.

We now divide the data set into two halves including the median in both data sets.

Lower half of the data set: 1, 2, 4, 5, 6.

Clearly, the median/middlemost term of the lower data set is 4. So we conclude that,

Lower Quartile = Q₁ = 4.

Upper half of the data set: 6, 7, 9, 11, 12.

The middlemost term of the upper data set is 9. So we conclude that,

Upper Quartile = Q₃ = 9.

Quartiles for Even Number of Data:

1. Arrange the data in ascending order.
2. Since N is even there are two terms lying exactly in the middle of the data values. They are the [N/2]^th term and the [(N/2)+1]^th term. The average of the two middlemost terms gives us the value of the median (second quartile).
3. Since N is even, the data values split into exactly two equal halves. The lower half and upper halves each contain (N/2) terms.
4. The median of the lower data set gives us the lower quartile and the median of the upper half gives us the value of Q₃.

Example 2:

Consider the following set of data values: 29, 32, 37, 39, 42, 44, 48, 53.

The two middlemost terms are the [N/2]^th term and the [(N/2)+1]^th term, that is the 4^th and 5^th terms. The average of these two terms gives us the median.

Median = Second Quartile = Q₂ = Average of 39 and 42 = (39+42)/2 = 40.5.

We now divide the data set into two halves. Each half contains exactly 4 data values.

Lower half of the data set: 29, 32, 37, 39.

Lower Quartile = Q₁ = Median of Lower Data Set = Average of 32 and 37 = 34.5.

Upper half of the data set: 42, 44, 48, 53.

Upper Quartile = Q₃ = Average of 44 and 48 = (44+48)/2 = 46.