Quartile Deviation Formulas

A quartile is a statistical tool that measures the variance of the given data by splitting the data into 4 equal parts and then comparing the results with the whole set of given observations and also articulates if any difference is found in the given data set. One of the most important uses of quartile is found in whisker and box plot.

Quartiles are the values that split the given data into 3 quarters. The three-quarters of quartiles are represented as:

  • The first quartile also known as Q1 is the median of the lower range of the given data set. It implies that about 25% of the numbers in the data set fall below the first quartile and about 75% of the data set falls above the first quartile.

  • The middle quartile also known as Q2 is considered as the median as it divides the data into 2 equal halves and also 50% of data falls below the Q2

  • The third quartile also known as  Q3 is the median of the upper range of the data set. It implies that about 75% of the numbers in the data set falls below Q3 and about 25% of the data falls above Q3 .

In this article we will study, quartile formula in statistics, quartile deviation formula, quartile deviation example, quartile formula for grouped data, quartile formula for ungrouped data, solved examples for quartile formulas etc.

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Quartile Definition

A quartile is nothing but a median that measures the spread of values above and below the mean by dividing the data into four parts. In other words, it is a value that splits the numbers into quarters.

For example, In {2, 4, 4, 5, 6, 7, 8} :

The value of Q1is equals to 4

The value of Q2  which is also the median is equals to 5

The value of Q3 is equals to 7


Quartile Formula in Statistics

Quartile splits the set of data into 4 quarters. 

  • The lower quartile also known as Q1 separates the lower portion of 25% of data from the highest portion of 75%.

  • The second quartile also known as Q2  is similar to the median as it divides the data into 2 equal halves.

  • The lower quartile also known as Q3 separates the upper portion of 25% off data from the lower portion of 75% of data.

When the given set of data is arranged in ascending order, and the number of n items are there in a data set, then  

The first quartile of the median of 25th percentile is calculated as

Q= (n + 1/4)th term

The middle quartile of the median of 50th percentile is calculated as

Q2 = (n + 1/2)th term

The third quartile of 75th percentile is calculated as

Q3 = (3(n + 1)/4)th term

Hence, the quartile formula is given as:

\[Q_{r} = l_{1} + \frac{r(\frac{N}{4})-c}{f}(l_{2}-l_{1})\]

In the above quartile formula,

Qr is the rth quartile

L1 is the lower limit

L2 is the upper limit

F is the frequency

C is the cumulative frequency of the above quartile class  

The upper quartile is calculated by rounding to the nearest whole number if the answer comes in the decimal numbers. The important use of the lower quartile and the upper quartile is that it measures the dispersion of the given data set. The dispersion is also known as the interquartile range and represented as IQR. 


Interquartile Range

The interquartile range is the difference between upper and lower quartiles of a given data set and is also known as midspread. It is the measure of statistical dispersion which is equal to the difference between 75th percentile and 35th percentile. If the value of Q3 and Q1 is given then the quartile formula for the range is given as  

IQR =  Q3 - Q1 


Quartile Formula for Grouped Data

The quartile formula for group data for calculating the values of Q1 ,Q2  and Q3 is

Q1 =  LQ1 + (n/4 - cf)/ fQ1)w

Q2 =  LQ2 + (n/2 - cf)/ fQ2)w

Q3 =  LQ3 + (3 n/4 - cf)/ fQ3)w

In the above quartile formula for grouped data 

  • LQ1 Denotes the lower limit of the class interval comprising Q

  • fQ1 Is the frequency of the class interval comprising Q1

  • W is the width of the class interval comprising Q1

  • Cf denotes the cumulative frequency upto but not comprising Q1 interval

Quartile Formula for Ungrouped Data

Quartile formula for ungrouped data for all the three quartiles is given as:

Q= (i *(n + 1/4))th value of observations.where i= 1,2,3

Q2 = (i *(n + 1/2))th value of observations.where i= 1,2,3

Q3 = (i*(3(n + 1)/4)th value of observations.where i= 1,2,3

For calculating the values of quartile for ungrouped data. The data has to be arranged in ascending order and then the quartile formula for ungrouped data is applied to calculate the values of Q1,,Q2 ,and Q3


Solved Examples For Quartile Formula

Below you can see solved examples for quartile formulas which will clear your concepts of quartile formulas in statistics.

  1. Calculate the Median, Lower Quartile, Upper Quartile and Interquartile Range of the Following Data Set of Runs 19, 21, 23, 20, 23, 27, 25, 24, 31.

Solution: 

The first step is to arrange the given data set in an ascending order:

19 ,20, 21, 23  23, 24, 25,27,and 31.

Now we will calculate the value of Q2 or median by using the following formula

Q2 = (n + 1/2)th term

Q2 = (9 + 1/2)th term

= 5th term i.e 23.

Value of upper quartile or Q3 is calculated by using the following formula

Q3 = (3(n + 1)/4)th term

Q3 = (3(9 + 1)/4)th term

30/4 or 7.5th term

7.5th term is calculated by taking the average of 7th and 8th terms

 =  (25 + 27)/2 = 26 

Hence, the value of Q3 is 26.

Value of upper quartile or Q1 is calculated by using the following formula

Q= (n + 1/4)th term

Q= (9 + 1/4)th term

= 2.5 th term

2.5th term is calculated by taking the average of 2nd and 3rd terms

 =  (20 + 21)/2 = 20.5 

Hence, the value of Q1 is 20.5.

The interquartile range is calculated by using the following formula:

IQR =  Q3 - Q1 

IQR = 26 - 20.5  

IQR = 5.5


  1. Calculate the Quartiles of the Following Data Set of Runs : 4, 6, 7, 8, 10, 23, 34.

Solution:

Here the numbers are arranged in the ascending order and the value of n = 7.

Q= (n + 1/4)th term

Q= (7 + 1/4)th term

= 2nd  term = 6

Hence, the value of Q1 is 6

Q2 = (n + 1/2)th term

Q2 = (7 + 1/2)th term

Q2 = (n + 1/2)th term

Q2 = 4th term = 8

Hence, the value of Qis 8.

Q3 = (3(n + 1)/4)th term

Q3 = (3(7 + 1)/4)th term

6th term = 23

Hence, the value of Q3 is 23


Quiz Time

1. The Value of the First Quartile is 23 and the Value of the Interquartile Range is 20 Then What Will Be the Value of the Third Quartile.

  1. 63

  2. 53

  3. 43

  4. 73


2. What is the Lower Quartile Among the Following?

25,14, 50,7 , 99, 41, 62

  1. 50.7

  2. 62

  3. 99

  4. 25

FAQ (Frequently Asked Questions)

1. What Are the Advantages and Disadvantages of Quartile Deviation?

Advantages of Quartile Deviation

  1. It is simply to understand and can be easily calculated.

  2. The value of quartile deviation is always fixed as it is a rigidly defined measure of dispersion.

  3. Quartile deviation is not much affected by extreme terms as 25% upper half and 25% of lower half values are left out.

  4. It relies on 50% of the observations.

  5. Quartile deviation can be ascertained in case of open distribution also.


Disadvantages of Quartile Deviation

  1. The value of quartile deviation is not much influenced by sampling fluctuations.

  2. It is not significant for further Mathematical process or algebraic operations.

  3. The calculation of quartile deviation is time-consuming as it is complex and needs some Mathematical understanding.

  4. It does not rely on all observations of a series.

  5. The calculation of quartile deviation in terms of continuous series becomes burdensome as it includes the application of the formula of interpolation.

2. What is Known as the Quartile Deviation?

Quartile deviation is the multiplication of half of the difference between upper and lower quartiles. Mathematically, quartile deviation is represented as;

Quartile Deviation Formula

[Q₃- Q₁]/2

In the above-given quartile deviation formula,

Q₃- It is the third quartile

Q₁- It is the first quartile

Quartile deviation explains the absolute measure of dispersion whereas the relative measure equivalent to quartile deviation is coefficient of quartile deviation which is acquired by using the set of formulas 

Coefficient of Quartile Deviation -  Q₃- Q₁/Q₃ + Q₁ * 100

A coefficient of quartile deviation can be explained for any distribution and also compares the degree of variations in different circumstances.