The Lesson
The lower quartile is the middle number between the smallest number and the median.It is the middle number of the lower half of a set of numbers. The lower quartile is denoted Q_{1}.
Understanding the Lower Quartile and the Quartiles
The lower quartile is the lowest of the three quartiles. The three quartiles divide a set of numbers, that are in numerical order, into four equal groups: The middle quartile is also known as the median. It is the middle number in the set. It divides the set in two halves: a lower half and an upper half.
 The lower quartile is the middle number of the bottom half. It divides the bottom half in two.
Finding the Lower Quartile
There are different methods for finding the lower quartile, which give different values.
The Moore and McCabe method excludes the median from the lower half. The lower quartile is the middle number of the lower half:

The Tukey method includes the median in the lower half. The lower quartile is the middle number of the lower half:
Note: When the lower half has an even number of numbers, the middle number is halfway between the middle two numbers. It is the mean of the middle two numbers:Q_{1} = (3 + 4) ÷ 2 = 3.5 
The Mendenhall and Sincich method uses a formula to find the position of the lower quartile. The formula is shown below:
In this formula, n is how many numbers there are in the set. In our example, n is 11.Position of Q_{1} = (n + 1) ÷ 4 = (11 + 1) ÷ 4 Position of Q_{1} = 12 ÷ 4 Position of Q_{1} = 3^{rd}The lower quartile is the 3^{rd} number in the set:
What's In a Name?
Quartile comes from the Latin word "quartus", meaning 'a fourth'. It comes from the same root as 'quarter'.Beware
Put Your Numbers in Order
The quartiles of a set of numbers divide the numbers into four equal groups when the numbers are in order. Imagine you were asked to find the lower quartile of the numbers below. Don't be tempted to jump right in.
3 5 4 1 2
Put the numbers in order and then find the lower quartile:
1 2 3 4 5