# Percentile

How to find the rank and the percentile of the value in the given data: formula, examples, and their solutions.

## Formula

The rank of a value shows
how small the number is in the data.

To find the percentile rank,
first find the rank of the value.

(rank) = n(smaller values) + (1/2)⋅n(same value)

n(smaller values): Number of the smaller values
n(same value): Number of the same values

(percentile) = (rank)/n

n: Number of the data

So, after finding the rank of the value
and the total number of the data,
you can find the percentile rank.

## Example 1

There are 6 numbers
that are smaller than 4.

And there's one 4.

So (rank) = 6 + (1/2)⋅1.

So the rank of 4 is 6.5th.

There are 20 numbers.
So n = 20.

(rank) = 6.5th
n = 20

So (percentile) = [6.5/20]⋅100.

Cancel 20
and reduce 100 to 5.

6.5⋅5 = 32.5

Round 32.5 to the nearest ones.

Then 32.5 → 33.

So 33rd percentile is the answer.

## Example 2

There are 12 numbers
that are smaller than 7.

And there are four 7.

So (rank) = 12 + (1/2)⋅4.

So the rank of 4 is 14th.

There are 20 numbers.
So n = 20.

(rank) = 14th
n = 20

So (percentile) = [14/20]⋅100.

Reduce 14 to 7
and reduce 20 to 10.

Cancel 10
and reduce 100 to 10.

7⋅10 = 70

So 70th percentile is the answer.

## Example 3

There are 20 - 2 = 18 scores
that are smaller than 5 points.

And there are two 5 points.

So (rank) = 18 + (1/2)⋅2.

Frequency table

So the rank of 5 points is 19th.

There are 20 numbers.
So n = 20.

(rank) = 19th
n = 20

So (percentile) = [19/20]⋅100.

Cancel 5
and reduce 100 to 20.

19⋅5 = 95

So 95th percentile is the answer.

## Example 4

There are 2 + 3 + 4 = 9 numbers
that are smaller than 75.

And there's one 75.

So (rank) = 9 + (1/2)⋅1.

Stem-and-leaf plot

So the rank of 75 is 9.5th.

There are 20 numbers.
So n = 20.

(rank) = 9.5th
n = 20

So (percentile) = [9.5/20]⋅100.

Cancel 20
and reduce 100 to 5.

9.5⋅5 = 47.5

Round 47.5 to the nearest ones.

Then 47.5 → 48.

So 48th percentile is the answer.