## Quantiles or Fractiles Uncovered (2020)

When the number of observations is sufficiently large, the principle by which a distribution is divided into two equal parts may be extended to divide the distribution into four, five, eight, ten, or hundred equal parts. The median, quartiles, deciles, and percentiles values are collectively called quantiles or fractiles. Let us start learning about Quantiles or Fractiles.

### Quantiles or Fractiles

#### Quartiles

These are the values that divide a distribution into four equal parts. There are three quartiles denoted by $Q_1, Q_2$, and $Q_3$. If $x_1,x_2,\cdots,x_n$ are $n$ observations on a variable $X$, and $x_{(1)}, x_{(2)}, \cdots, x_{(n)}$ is their array then $r$th quartile $Q_r$ is the values of $X$, such that $\frac{r}{4}$ of the observations is less than that value of $X$ and $\frac{4-r}{4}$ of the observations is greater.

The $Q_1$ is the value of $X$ such that $\frac{1}{4}$ of the observations is less than the value of $X$ and $\frac{4-1}{4}$ of the observations is greater, the $Q_3$ is the value of $X$, such that $\frac{3}{4}$ of the observations is less than that of $X$ and $\frac{4-3}{4}$ of the observations is greater.

#### Deciles

These are the values that divide a distribution into ten equal parts. There are 9 deciles $D_1, D_2, \cdots, D_9$.

#### Percentiles

These are the values that divide a distribution into a hundred equal parts. There are 99 percentiles denoted as $P_1,P_2,\cdots, P_{99}$.

The median, quartiles, deciles, percentiles, and other partition values are collectively called quantiles or fractiles. All quantiles are percentages. For example, $P_{50}, Q_2$, and $D_5$ are also median.

\begin{align*}
Q_2 &= D_5 = P_{50}\\
Q_1 &= P_{25} = D_{2.5}\\
Q_3 &= P_{75}=D_{7.5}
\end{align*}
The $r$th quantile, $k$th decile, and $j$th percentile are located in the array by the following relation:
For ungrouped Date

\begin{align}
Q_r &=\frac{r(n+1)}{4}\text{th value in the distribution and } r=1,2,3\\
D_k &=\frac{k(n+1)}{10}\text{th value in the distribution and } k=1,2,\cdots, 9\\
P_j &=\frac{j(n+1)}{100}\text{th value in the distribution and } k=1,2,\cdots, 99
\end{align}
For grouped Data
\begin{align}
Q_r&= l+\frac{h}{f}\left(\frac{rn}{4}-c\right)\\
D_k&= l+\frac{h}{f}\left(\frac{kn}{10}-c\right)\\
P_j&= l+\frac{h}{f}\left(\frac{jn}{100}-c\right)
\end{align}

A procedure for obtaining percentile (quartiles, deciles) of a data set of size $n$ is as follows:

Step 1: Arrange the data in ascending/ descending order.
Step 2: Compute an index $i$ as follows: $i=\frac{p}{100} (n+1)$th (in case of odd observation).

• If $i$ is an integer, the $p$th percentile is the average of the $i$th and $(i+1)$th data values.
• if $i$ is not an integer then round $i$ up to the nearest integer and take the value at that position or use some mathematics to locate the value of percentile between $i$th and $(i+1)$th value.

Percentile Example:

Consider the following (sorted) data values: 380, 600, 690, 890, 1050, 1100, 1200, 1900, 890000.

For the $p=10$th percentile, $i=\frac{p}{100} (n+1) =\frac{10}{100} (9+1)= 1$. So the 10th percentile is the first sorted value or 380.

For the $p=75$ percentile, $i=\frac{p}{100} (n+1)= \frac{75}{100}(9+1) = 7.5$

To get the actual value we need to compute 7th value + (8th value – 7th value) $\times 0.5$. That is, $1200 + (1900-1200)\times 0.5 = 1200+350 = 1550$.