Introduction to Quantiles and Quartiles
Quantiles are the techniques used to divide the data into different equal parts. For example, quantiles divide the data into four equal parts. Quartile comes from quarter which means 4th part. Deciles divide the data into ten equal parts and they come from deca means the 10th part. Percentiles divide the data into hundred parts and it comes to percent which means the 100th part.
Table of Contents
Therefore, quartiles, deciles, and percentiles are used to divide the data into 4, 10, and 100 parts respectively. The quantiles, deciles, and percentiles are collectively called quantiles.
Quartiles are the rules which divide the data into four equal parts. When we divide any data into four equal parts then we cut it at e equidistant points. Therequartiles ($Q_1, Q_2$, and $Q_3$) as quartiles divide the data into four equal parts so divide the number of observations by four for each quartile.
Quartiles for Ungroup Data
\begin{align*}
Q_1 &= \left(\frac{n+1}{4}\right)th \text{ value is the} \frac{1}{4} \text{ part}\\
Q_2 &= \left(\frac{2(n+1)}{4}\right)th \text{ value is the} \frac{2}{4} \text{ part}\\
\left(\frac{3(n+1)}{4}\right)th \text{ value is the} \frac{3}{4} \text{ part}
\end{align*}
The following ungroup data has 96 observations $(n=96)$
| 22 | 22 | 25 | 25 | 30 | 30 | 30 | 31 | 31 | 33 | 36 | 39 |
| 40 | 40 | 42 | 42 | 48 | 48 | 50 | 51 | 52 | 55 | 57 | 59 |
| 81 | 86 | 89 | 89 | 90 | 91 | 91 | 91 | 92 | 93 | 93 | 93 |
| 93 | 94 | 94 | 94 | 95 | 96 | 96 | 96 | 97 | 97 | 98 | 98 |
| 99 | 99 | 99 | 100 | 100 | 100 | 101 | 101 | 102 | 102 | 102 | 102 |
| 102 | 103 | 103 | 104 | 104 | 104 | 105 | 106 | 106 | 106 | 107 | 108 |
| 108 | 108 | 109 | 109 | 109 | 110 | 111 | 112 | 112 | 113 | 113 | 113 |
| 113 | 114 | 115 | 116 | 116 | 117 | 117 | 117 | 118 | 118 | 119 | 121 |
The first, second, and third quartiles of the above data set are:
\begin{align*}
Q_1 &= \left(\frac{n}{4}\right)th \text{ position } = \left(\frac{96}{4} = 24th \text{ value} = 59\\
Q_2 &= \left(\frac{2\times 96}{4}\right) = 48th \text{position} = 98\\
Q_3 &= \left(\frac{3\times n}{4}\right)th = \left(\frac{3\times 96}{}\right)th \text{ position} = 72th \text{ position} = 108
\end{align*}
Note that the above data is already sorted. If data is not sorted, first we need to arrange/sort the data in ascending order.
Quartiles for Gruoped Data
For the following grouped data one can also compute the quantiles, hence the quartiles.
| Classes | f | x | C.B. | CF |
|---|---|---|---|---|
| 65-84 | 9 | 74.5 | 64.5-84.5 | 9 |
| 85-104 | 10 | 94.5 | 84.5-104.5 | 19 |
| 105-124 | 17 | 114.5 | 104.4.5-124.5 | 36 |
| 125-144 | 10 | 134.5 | 124.5-144.5 | 46 |
| 145-164 | 5 | 154.5 | 144.5-164.5 | 51 |
| 165-184 | 4 | 174.5 | 164.5-184.4 | 55 |
| 185-204 | 5 | 194.5 | 184.5-204.5 | 60 |
| Total | 60 |
From the above-grouped data, we have 60 observations $(n=60)= \sum\limits_{i=1}^n = f_i = \Sigma f = 60$. The three quartile will be
\begin{align*}
\frac{n}{4} &= \left(\frac{60}{4}\right)th = 15th \text{ value}\\
Q_1 &= l + \frac{h}{f}\left(\frac{n}{4} – CF\right) = 84.5 + \frac{20}{10}(15-9) = 96.5\\
\frac{2n}{4} &= \left(\frac{2\times 60}{4} \right) = 30th \text{ value}\\
Q_2 &= l + \frac{h}{f}\left(\frac{2n}{4} – CF\right) = 104.5 + \frac{20}{17}(30-19) = 117.44\\
\frac{3n}{4} &= \left(\frac{3\times 60}{4} \right) = 45th \text{ value}\\
Q_3 &= l + \frac{h}{f}\left(\frac{3n}{4} – CF\right) = 124.5 + \frac{20}{17}(45-36) = 142.5\\
\end{align*}
