MODE

 The mode is the most number of occurence of a  value in a data. That is it appears 

most frequently in a data. An observation sometimes  have  no mode ,one mode, 

two mode or more than two mode . There are different types of mode. If an 

observation has one mode it is called unimodal. An observation with two mode is 

called bimodal. An observations with three mode,  it is called  trimodal. If it is three 

or more , it is multimodal .It is one of the measure of central tendency. 

Other measures of central tendency are mean and median . 

There is a relation between mean, median and mode.

                  i. e,  MODE = 3 MEDIAN - 2 MEAN. 

There are two types of data : ungrouped and grouped

Un Grouped data:

In a grouped data or in an individual series ,Mode is defined as the most repeating 

value in an observation or the number that occurs highest number of times

Example:

Find the mode of 

     6, 3, 4 ,5 ,6 , 7, 2,10

Solution:

  Most repeating value in the observation 6, 3, 4 ,5 ,6, 7 , 2, 10 is 6  (repeated 

twice).

                Therefore Mode = 6

Example:

Find the mode of 

   2, 3, 2, 4 , 8, 2, 5, 2, 3 , 7, 3

Solution;

   In this observations there are two numbers repeating 3 times

          So there are two modes: 2 and 3

Example

In a class test in English 10 students scored 75 marks 12 students scored 60 marks 8 

scored 40 marks and 3 scored 30 marks the mode for their score is:

solution:

The mode of a set of numbers is the observation that appears most often or is most 

frequent.

From given data:

$10$ students scored$=75$ marks

$12$ students scored $=60$ marks

$8$ students scored $=40$ marks

$3$ student scored $=30$ marks

     Hence, in the set of numbers $60$ appear most often

      So, $60$ is the mode .

Grouped data or Continous series:

    It is not possible to find the mode of grouped data by looking at the frequencies 

in the distribution table. For finding the mode for a grouped data we have a 

formula. It is given by

                         

$Mode=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

Example

Find mode of the following data :

Class-interval	Frequency
80-85	33
85-90	27
90-95	85
95-100	155
100-105	110
105-110	45
110-115	15

Solution; 

Here maximum frequency $=155$
whose class is $95-100$
$\therefore i=95,f_1=155,f_0=85,f_2=110,h=5$

$\because Mode=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$
$=95+\left(\frac{155-85}{2\times 155-85-110}\right)\times 5$

$=95+\frac{70\times 5}{310-195}$

$=95+\frac{70}{23}$

$=95+3.04=98.04$
     

     Thus, required mode $=98.04$

 Example;

Find mode of the following data :

Class-interval	Frequency
10 - 20	4
20 -30	6
30 -40	12
40 -50	8
50 -60	5
60 -70	2
70 -80	3

Solution; 

Here maximum frequency $=\; 12$
whose class is 30 -40
$\therefore l=30,f_1=12,f_0=6,f_2=8,h=10$

$\because Mode=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

            = 30 + (12- 6) 10 / 2*12 - 6 -8

            =  30  +(6*10) / 24-14

            = 30 + 60/10

            = 36

Thus, required mode $=\; 36$

The empirical relationship between Mean  Median and Mode:

Empirical relationship between mean median and mode for a moderately skewed 

distribution can be given as:

  

             MODE = 3 MEDIAN - 2 MEAN. 

 Example:

In a moderately skewed distribution, the median is 20 and the mean is 22.5. Using 

these values, find the approximate value of the mode.

Solution:

Given,

    

     Mean = 22.5

   

     Median = 20

  

     Mode = x

     Now, using the relationship between mean mode and median we get,

              

     MODE = 3 MEDIAN - 2 MEAN. 

     Mode= 3 Median - 2 Mean

     So,

                  x = 3 * 20 – 2 *22.5

                   x = 60 -45

       

                 ∴ x = 15

               So, Mode = 15.

 

Read More:

                 

•      Mean formula
•       Median formula
•       Algebraic identities
•       Basic derivative formula
•       Basic integral formula
•       Basic trigonometric formula
•       Basic rules of exponents and powers