A quartile divides the set of observation into 4 equal parts. The middle term, between the median and first term is known as the first or Lower Quartile and is written as Q_{1}. Similarly, the value of mid term that lies between the last term and the median is known as the third or upper quartile and is denoted as Q_{3}. Second Quartile is the median and is written as Q_{2}.

When the set of observation is arranged in an ascending order, then the 25th percentile is given as:

\[\large Q_{1}=\left(\frac{n+1}{4}\right)^{th}Term\]

The second quartile or the 50th percentile or the Median is given as:

\[\large Q_{2}=\left(\frac{n+1}{2}\right)^{th}Term\]

The third Quartile of the 75th Percentile (Q3) is given as:

\[\large Q_{3}=\left(\frac{3(n+1)}{4}\right)^{th}Term\] The Upper quartile is given by rounding to the nearest whole integer if the solution is coming in decimal number. The major use of the lower and upper quartile helps is that it helps us measure the dispersion in the set of the data given. The dispersion is also called “inter quartile range”, denoted as IQR, inter quartile range is the difference between lower and upper quartile.

\[\large IQR=Upper\;Quartile-Lower\;Quartile\]

To find the quartile we first need to arrange the values in ascending order. Then we need to put the formula to use. Let’s solve one example to make it clear to you:

### Solved example

**Question:** Find the median, lower quartile, upper quartile and inter-quartile range of the following data set of scores: 19, 21, 23, 20, 23, 27, 25, 24, 31 ?

**Solution:**

First, lets arrange of the values in an ascending order: 19, 20, 21, 23, 23, 24, 25, 27, 31

Now let’s calculate the Median,

$Q_{2}=\left(\frac{n+1}{2}\right)^{th}Term$

$Q_{2}=\left(\frac{9+1}{2}\right)^{th}Term$

\(=5^{th}\;Term\; = 23\)Lower Quartile:

$Q_{1}=\left(\frac{n+1}{4}\right)^{th}Term$

$=\left(\frac{9+1}{4}\right)^{th}\;Term=2.5^{th}\;Term$

Upper Quartile:

$Q_{3}=\left(\frac{3(n+1)}{4}\right)^{th}Term$

$Q_{3}=\left(\frac{3(9+1)}{4}\right)^{th}Term$

$\frac{30}{4}=7.5^{th}\;Term$

Average of 2nd and 3rd terms

= (20 + 21)/2 = 20.5 = Lower Quartile

Average of 7th and 8th terms

= (25 + 27)/2 = 26 = Upper Quartile

IQR = Upper quartile – Lower quartile

= 26 – 20.5

= 5.5

More topics in Quartile Formula | |

Interquartile Range Formula |