Percentiles: Measure of relative standing of an observation within data

Percentiles are measure of relative standing of an observation within a data. Percentiles divides 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 etc.

The pth 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) percent of the values are greater Y(p) . The 5th percentile is denoted by P5 , the 10th by P10 and 95th by P95 .

Percentiles for the ungrouped data

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

  1. Order the observation
  2. For the mth 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 Kth 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  P10 is 3rd observation in sorted data is 88, means that 10 percent of the observations in data set are less than 88.

\[P_{95}=\frac{95 \times 30}{100}=28.5\]

29th observation is our 95th percentile i.e. P95=107.

Percentiles for the Grouped data

The mth percentile (measure of relative standing of an observation) for 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 mth percentile group.

l     is the lower class boundary of the class containing the mth percentile
h   is the width of the class containing Pm
f    is the frequency of the class containing
n   is the total number of frequencies Pm
c    is the cumulative frequency of the class immediately preceding to the class containing Pm

Note that 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 25th and 75th percentiles are the lower (Q1) and upper quartiles (Q3) respectively. The quartiles, deciles and percentiles are also called quantiles or fractiles.

Deciles, Percentiles for Grouped data

Measure of relative standing of an observation in Grouped Data

Example: For the following grouped data compute P10 , P25 , P50 , and P95 given below.Solution:

  1. Locate the 10th percentile (lower deciles i.e. D1)by $\frac{10 \times n}{100}=\frac{10 \times 3o}{100}=3$ observation.
    so, P10 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. Q1)  by $\frac{10 \times n}{100}=\frac{25 \times 3o}{100}=7.5$ observation.
    so, P25 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 $\frac{95 \times n}{100}=\frac{95 \times 3o}{100}=28.5$th observation.
    so, P95 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 cumulative frequency function.

Further Reading: https://en.wikipedia.org/wiki/Percentile

 Download pdf file: 
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