# Measure of Position

## Quantiles or Fractiles

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, $1200 + (1900-1200)\times 0.5 = 1200+350 = 1550$.

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## Quartiles in Statistics: Relative Measure of Observation

#### Quartiles in Statistics

Like Percentiles and Deciles, Quartiles is a type of Quantile, which is a measure of the relative standing of observation within the data set. The Quartiles values are three points that divide the data into four equal parts each group comprising a quarter of the data (the first quartile $Q_1$, second quartile $Q_2$ (also median), and the third quartile $Q_3$) in the order statistics. The first quartile, (also known as the lower quartile) is the value of order statistic that exceeds 1/4 of the observations and less than the remaining 3/4 observations. The third quartile known as the upper quartile is the value in the order statistic that exceeds 3/4 of the observations and is less than the remaining 1/4 observations, while the second quartile is the median.

#### Quartiles in Statistics for Ungrouped Data

For ungrouped data, the quartiles are calculated by splitting the order statistic at the median and then calculating the median of the two halves. If $n$ is odd, the median can be included on both sides.

Example: Find the $Q_1, Q_2$ and $Q_3$ for the following ungrouped data set 88.03, 94.50, 94.90, 95.05, 84.60.Solution: We split the order statistic at the median and calculated the median of two halves. Since $n$ is odd, we can include the median in both halves. The order statistic is 84.60, 88.03, 94.50, 94.90, 95.05.

\begin{align*}
Q_2&=median=Y_{(\frac{n+1}{2})}=Y_{(3)}\\
&=94.50  (\text{the third observation})\\
Q_1&=\text{Median of the first three value}=Y_{(\frac{3+1}{2})}\\&=Y_{(2)}=88.03 (\text{the second observation})\\
Q_3&=\text{Median of the last three values}=Y_{(\frac{3+5}{2})}\\
&=Y_{(4)}=94.90 (\text{the fourth observation})
\end{align*}

#### Quartiles in Statistics for Grouped Data

For the grouped data (in ascending order) the quartiles are calculated as:
\begin{align*}
Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\
Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\
Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)
\end{align*}
where
$l$    is the lower class boundary of the class containing the $Q_1, Q_2$ or $Q_3$.
$h$    is the width of the class containing the $Q_1, Q_2$ or $Q_3$.
$f$    is the frequency of the class containing the $Q_1, Q_2$ or $Q_3$.
$c$    is the cumulative frequency of the class immediately preceding the class containing $Q_1, Q_2$ or $Q_3, \left[\frac{n}{4},\frac{2n}{4} \text{or} \frac{3n}{4}\right]$ are used to locate $Q_1, Q_2$ or $Q_3$ group.

Quartiles in Statistics Example: Find the quartiles for the following grouped data

Solution: To locate the class containing $Q_1$, find $\frac{n}{4}$th observation which is here $\frac{30}{4}$th observation i.e. 7.5th observation. Note that the 7.5th observation falls in the group ($Q_1$ group) 90.5–95.5.
\begin{align*}
Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\
&=90.5+\frac{5}{4}(7.5-6)=90.3750
\end{align*}

For $Q_2$, the $\frac{2n}{4}$th observation=$\frac{2 \times 30}{4}$th observation = 15th observation falls in the group 95.5–100.5.
\begin{align*}
Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\
&=95.5+\frac{5}{10}(15-10)=98
\end{align*}

For $Q_3$, the $\frac{3n}{4}$th observation=$\frac{3\times 30}{4}$th = 22.5th observation. So
\begin{align*}
Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)\\
&=100.5+\frac{5}{6}(22.5-20)=102.5833
\end{align*}

Reference:

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## Percentiles: Measure of Relative Standing

Percentiles are a measure of the relative standing of observation within a data. Percentiles divide a set of observations into 100 equal parts, and percentile scores are frequently used to report results from national standardized tests such as NAT, GAT, and GRE, etc.

The $p$th percentile is the value $Y_{(p)}$ in order statistic such that $p$ percent of the values are less than the value $Y_{(p)}$ and $(100-p)$ (100-p) percent of the values are greater $Y_{(p)}$. The 5th percentile is denoted by $P_5$, the 10th by $P_{10}$ and 95th by $P_{95}$.

### Percentiles for the Ungrouped data

To calculate percentiles (a measure of the relative standing of an observation) for the ungrouped data, adopt the following procedure:

1. Order the observation
2. For the $m$th percentile, determine the product $\frac{m.n}{100}$. If $\frac{m.n}{100}$ is not an integer, round it up and find the corresponding ordered value and if $\frac{m.n}{100}$ is an integer, say k, then calculate the mean of the $K$th and $(k+1)$th ordered observations.

Example: For the following height data collected from students find the 10th and 95th percentiles. 91, 89, 88, 87, 89, 91, 87, 92, 90, 98, 95, 97, 96, 100, 101, 96, 98, 99, 98, 100, 102, 99, 101, 105, 103, 107, 105, 106, 107, 112.

Solution: The ordered observations of the data are 87, 87, 88, 89, 89, 90, 91, 91, 92, 95, 96, 96, 97, 98, 98, 98, 99, 99, 100, 100, 101, 101, 102, 103, 105, 105, 106, 107, 107, 112.

$P_{10}= \frac{10 \times 30}{100}=3$

So the 10th percentile i.e. $P_{10}$ is the 3rd observation in sorted data is 88, which means that 10 percent of the observations in the data set are less than 88.

$P_{95}=\frac{95 \times 30}{100}=28.5$

The 29th observation is our 95th Percnetile i.e., $P_{95}=107$

### Percentiles for the Frequency Distribution Table (Grouped data)

The $m$th percentile (a measure of the relative standing of an observation) for the Frequency Distribution Table (grouped data) is

$P_m=l+\frac{h}{f}\left(\frac{m.n}{100}-c\right)$

Like median, $\frac{m.n}{100}$ is used to locate the $m$th percentile group.

$l$    is the lower class boundary of the class containing the $m$th percentile
$h$   is the width of the class containing $P_m$
$f$    is the frequency of the class containing
$n$   is the total number of frequencies $P_m$
$c$    is the cumulative frequency of the class immediately preceding the class containing $P_m$

Note that the 50th percentile is the median by definition as half of the values in the data are smaller than the median and half of the values are larger than the median. Similarly, the 25th and 75th percentiles are the lower ($Q_1$) and upper quartiles ($Q_3$) respectively. The quartiles, deciles, and percentiles are also called quantiles or fractiles.

Example: For the following grouped data compute $P_{10}$, $P_{25}$, $P_{50}$, and $P_{95}$ given below.Solution:

1. Locate the 10th percentile (lower deciles i.e. $D_1$)by $\frac{10 \times n}{100}=\frac{10 \times 3o}{100}=3$ observation.
so, $P_{10}$ group is 85.5–90.5 containing the 3rd observation
\begin{align*}
P_{10}&=l+\frac{h}{f}\left(\frac{10 n}{100}-c\right)\\
&=85.5+\frac{5}{6}(3-0)\\
&=85.5+2.5=88
\end{align*}
2. Locate the 25th percentile (lower quartiles i.e. $Q_1$)  by $\frac{10 \times n}{100}=\frac{25 \times 3o}{100}=7.5$ observation.
so, $P_{25}$ group is 90.5–95.5 containing the 7.5th observation
\begin{align*}
P_{25}&=l+\frac{h}{f}\left(\frac{25 n}{100}-c\right)\\
&=90.5+\frac{5}{4}(7.5-6)\\
&=90.5+1.875=92.375
\end{align*}
3. Locate the 50th percentile (Median i.e. 2nd quartiles, 5th deciles) by $\frac{50 \times n}{100}=\frac{50 \times 3o}{100}=15$ observation.
so, P50 group is 95.5–100.5 containing the 15th observation
\begin{align*}
P_{50}&=l+\frac{h}{f}\left(\frac{50 n}{100}-c\right)\\
&=95.5+\frac{5}{10}(15-10)\\
&=95.5+2.5=98
\end{align*}
4. Locate the 95th percentile by $\frac{95 \times n}{100}=\frac{95 \times 30}{100}=28.5$th observation.
so, $P_{95}$ group is 105.5–110.5 containing the 3rd observation
\begin{align*}
P_{95}&=l+\frac{h}{f}\left(\frac{95 n}{100}-c\right)\\
&=105.5+\frac{5}{3}(28.5-26)\\
&=105.5+4.1667=109.6667
\end{align*}

The percentiles and quartiles may be read directly from the graphs of the cumulative frequency function.