Percentiles: Measure of relative standing of an observation within data

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, 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 $P_5$, 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 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 the 3rd observation in sorted data is 88, which means that 10 percent of the observations in the data set are less than 88.

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

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

Percentiles for the Grouped data

The mth percentile (a measure of the 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 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 to 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, 25th and 75th percentiles are the lower (Q₁) and upper quartiles (Q₃) respectively. The quartiles, deciles, and percentiles are also called quantiles or fractiles.

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