Quartiles is the four equal part of a population which is divided according to a particular variable. In statistics, Quartile divides the set of observation into 4 equal parts like how the median divides the set of observation into 2 equal parts, when arranged in the numerical order.
There are three points which divide the set of observation into 4 equal parts(as shown in the figure below). The three points are:
  1. First or Lower quartile (Q1)
  2. Second quartile or Median (Q2)
  3. Third quartile or Upper quartile (Q3)

Quartiles

The middle term value between the first term and the median is known as First or Lower quartile and is denoted as Q1. The formula is given as:

Quartile Formula
The median itself is known as Second quartile and is denoted as Q2. The formula is given as:

Quartile Formulas
The middle term value between the last term and the median is known as Third or Upper quartile and is denoted as Q3. The formula is given as:

Quartiles Formula
The half the difference between the upper and lower quartile in a set of observation is called as Quartile deviation. It is useful as it is not connected by extremely high or extremely low scores.

The formula of Quartile deviation is given as follows:

Quartile Deviation
Let us find the quartile for the following data:
4, 7, 3, 8, 20, 18, 5

First let us arrange the given data in ascending order
3, 4, 5, 7, 8, 18, 20
Number of Observation (n) = 7
Lower quartile, $Q_{1}$ = $\left ( \frac{n+1}{4} \right )^{th}$ term
= $(\frac{7+1}{4})^{th}$ term
= 2nd term
i.e. 4
Second Quartile, $Q_{2}$ = $\left ( \frac{n+1}{2} \right )^{th}$ term
= $(\frac{7+1}{2})^{th}$ term
= 4th term
i.e. 7
Upper Quartile, $Q_3$ = $\left ( \frac{3(n+1)}{4} \right )^{th}$ term
= $\left ( \frac{3(7+1)}{4} \right )^{th}$ term
= $\left ( \frac{3(8)}{4} \right )^{th}$ term
= 6th term
i.e. 18
The differences between the upper and lower quartile is called as Interquartile range. Interquartile range is also known as the midspread and is a measure of dispersion. It is the difference between the third quartile and the first quartile and is used to find outliers in a given data set. Through inter quartile range, we can see how the data will be spread about the median.

Interquartile Range

This range helps us to find the measure of dispersion in the set of observation. Its is denoted as 'IQR' and the formula is given as:

Interquartile Range Formula → Read More
The income quartile method helps to know the even distribution within the each income group. The income quartile are divided to examine the income distribution household. The major feature of the households is that they fall under different income quartiles. Households are divided into eight income classes. That is, 4 income quartiles in each of rural and urban areas. And, on the basis of a income survey data, factor incomes in agriculture and non agriculture are distributed to each quartile.
Nine values of a variables forms a decile and its distribution is divided into ten parts with equal frequencies. The tenth part of any given distribution is called as decile. Here, the set of data should be arranged in ascending or descending order, so that we can find out the values before dividing.

Each deciles are defined and denoted as stated below:

  1. First decile: It covers 10% of the total population and is denoted as D1
  2. Second decile: It covers 20% of the total population and is denoted as D2
  3. Third decile: It covers 30%of the total population and is denoted as D3
  4. Forth decile: It covers 40% of the total population and is denoted as D4
  5. Fifth decile: It covers 50% of the total population abd is denoted as D5
  6. Sixth decile: It covers 60% of the totol population and is denoted as D6
  7. Seventh decile: It covers 70% of the total population and is denoted as D7
  8. Eight decile: It covers 80% of the total population and is denoted as D8
  9. Ninth decile: It covers 90% of the total population and is denoted as D9
By this. we understand that the deciles can conclude the values for 10% to 90% of the data which helps us to know the distribution of data and also the range.
Percentiles are also like the quartiles. The main difference between them is that quartiles divides the data into 4 parts whereas, the percentiles divides the data into 100 equal parts.

It helps us to know at what percent of the total frequency scored below one particular measure.

There are two ways to calculate the percentiles.

1. When the percentile rank score 'x' is included, out of a set of 'n' scores. The formula for this is given as

PercentilesHere,
B = number of scores below x
E = number of scores equal to x
n = number of scores

2. When the percentile rank score 'x' is not included, out of a set of 'n' scores. The formula for this is given as

Percentile
Given below are few examples problems based on quartiles.

Solved Examples

Question 1: For the following data of scores: 12, 22, 24, 27, 16, 19, 15, find first quartile, second quartile, third quartile and Interquartile Range.
Solution:
Let us arrange the values in ascending order:12, 15, 16, 19, 22, 24, 27
Given n = 7
Lower quartile = $\left ( \frac{n+1}{4} \right )^{th}$ term = $\left ( \frac{7+1}{4} \right )^{th}$ term = $\left ( \frac{8}{4} \right )^{th}$ term = 2nd term
i.e. 15
Second Quartile = $\left ( \frac{n+1}{2} \right )^{th}$ term = $\left ( \frac{7+1}{2} \right )^{th}$ term = $\left ( \frac{8}{2} \right )^{th}$ term = 4th term i.e. 19
Lower Quartile = $\left ( \frac{3(n+1)}{4} \right )^{th}$ term = $\left ( \frac{3(7+1)}{4} \right )^{th}$ term = $\left ( \frac{3(8)}{4} \right )^{th}$ term = 6th term i.e. 24
Inter-quartile range = Upper quartile - Lower Quartile = 15 - 24 = -9

Question 2: If Ravi graduated 26th out of a class of 120 students, then the other 94 students were ranked below Ravi. Find the Percentile rank of Ravi.
Solution:
Here :B = 94E = 1n = 120Percentile Rank = $\frac{94+0.5(1)}{120}$$ \times 100$

= $\frac{94.5}{120}$$ \times 100$

= 79th percentile