Measures of Central tendency: Mode

A commonly used measure of central tendency, apart from Arithmetic Mean and Median, is Mode. The fundamental definition of Mode is the most frequently occurring value of the variable, i.e. that value of the variable which occurs the maximum number of times in a data set. If all values of the variable occur equal number of times, then mode is undefined. However, if more than one value of the variable has maximum frequency, then mode is defined, but not unique. It finds use in the field of business. For example, most shoe stores prefer to stock the “most popular size”, i.e. the most common size which is purchased most usually by the customers. Strictly speaking, the above definition of mode is applicable to discrete variables only. For calculating mode in case of a frequency distribution of a continuous variable we will use a formula. Mode is usually denoted by the symbol $\breve{x}$ .

Mode for Non-frequency Type Data (Discrete Variables only):

For non-frequency type data, i.e. for raw data, mode can be calculated merely by inspecting the values in the data set. Mode ( $\breve{x}$ ) is that value of the variable which the maximum number of times in the given data set.

Example: A sample of 10 shoppers at a shopping mall were selected and each of their shoe size was recorded: 8, 6, 8, 7, 9, 7, 7, 5, 6, 7. Determine the mode.

Solution:

Here, the value 7 occurs the maximum number of times. So, the mode is 7.

$\therefore \breve{x} =7$

Mode for Frequency Type Data:

(a) Simple Frequency Distribution (Discrete Variables only):

For simple frequency distribution, i.e. a frequency distribution without class intervals, the mode can be found, merely by inspection. Mode is that value for which the corresponding frequency is the highest.

Example: The following table shows the frequency distribution of the number of children per family for a sample of 80 families. Find out the mode of this distribution.

Number of Children	Number of Families
0	10
1	24
2	36
3	10
Total	80

Solution:

Here, the value of the variable with maximum frequency is 2 with frequency 36. In this data set, 2 has occurred maximum number of times when compared to the other values.

$\therefore$ the required mode is, $\breve{x} =2$

**(b) Grouped Frequency Distribution (Discrete and Continuous Variables):**

When the data under consideration is grouped into class intervals, mode can be determined by using a formula, but only if the classes are of equal width.

First, we identify the modal class. The modal class is defined as that class interval which has the highest frequency. Then we can calculate the mode using the following formula:

Mode ( $\breve{x} )= x_{l} + \dfrac{f_{0}-f_{-1}}{2f_{0}-f_{1}-f_{-1}} \times w$

where, $x_{l}$ is the lower class boundary of the modal class.

$f_{0}$ is the frequency of the modal class.

$f_{1}$ is the frequency corresponding to the class next to the modal class.

$f_{-1}$ is the frequency of the class previous to the modal class.

$w$ is the equal class width.

Example: The following frequency distribution table shows the monthly wage-distribution of 130 workers of a factory. Obtain the mode of the following distribution.

Monthly Wage (in Rupees)	Number of Workers
1500-1700	25
1700-1900	30
1900-2100	37
2100-2300	27
2300-2500	11
Total	130

Solution:

Here, the maximum frequency is 37. The class corresponding to which the frequency is 37 is 1900-2100. So, 1900-2100 is the modal class.

Here, $x_{l}=1900$ is the lower class boundary of the modal class.

$f_{0}=37$ is the frequency of the modal class.

$f_{1}=27$ is the frequency corresponding to the class next to the modal class, i.e. 2100-2300.

$f_{-1}=30$ is the frequency of the class previous to the modal class, i.e. 1700-1900.

$w=200$ is the equal class width.

$\therefore$ the required mode is:

$\breve{x} = 1900 + \dfrac{37-30}{2 \times 37 -30-27} \times 200$

$= 1900+\dfrac{1400}{17}=1982.35$ (Rs.)

Note: In this example, we used a frequency distribution for which the class boundaries were given. If the class intervals are given instead of class boundaries, one has to convert the class intervals to class boundaries before one can start calculating the mode of the frequency distribution.

An Important Property of Mode:

If two variables $x$ and $y$ have one to one correspondence described by the function $y=f(x)$ , then their modes are also related by the same relationship:

$y=f(\breve{x})$

What this basically means is that the value which has the highest frequency in the $x$ -series, i.e. $\breve{x}$ , will correspond to $f(\breve{x})$ in the $f(x)$ -series or the $y$ -series. For this property to0 hold, however, we must have one to one correspondence between $x$ and $y$ which means that there exists a unique value of $y$ corresponding to each value of $x$ .

Note: In particular, if $x$ and $y$ are linearly related as $y=a+bx$ , $a,b$ being two constants, then the modes of $x$ and $y$ will be linearly related as

$\breve{y}=a+b\breve{x}$

Merits and Demerits of Mode:

Merits:

Mode is easy to understand and easy to calculate.
In some cases, it can be determined merely by inspection.
Mode is not affected by the presence of extreme values.

Demerits:

Mode is not directly based upon all observations.
It cannot be algebraically treated.
If class widths are unequal for a particular frequency distribution, mode cannot be determined for that frequency distribution.
Mode is may not be unique. A frequency distribution may be bimodal or even multimodal, i.e. there may be more than one value which has maximum frequency.

Exercise:

1. Find the mode of the following data:

9, 11, 8, 11, 16, 9, 11, 5, 3, 11, 17, 8.

2. Find the mode of the following data.

Class	0-10	10-20	20-30	30-40	40-50
Frequency	5	12	20	9	4

3. Find the mode from the following frequency distribution.

Number	8	9	10	11	12	13	14	15	16
Frequency	3	8	12	15	14	17	12	8	6

4. The following table shows the frequency distribution of heights of 50 boys. Find the mode.

Height	120	121	122	123	124
Frequency	5	8	18	10	9

5. The following table shows the frequency distribution of the weekly wages of workers in a factory. Find the modal wage of the workers in this factory.

Weekly wages	No. Of workers
50-55	5
55-60	20
60-65	10
65-70	10
70-75	9
75-80	6
80-85	12
85-90	8

6. Frequency distribution of IQ for 309 6-year old children is give below. Calculate the mode.

Class Intervals	Frequency
160-169	2
150-159	3
140-149	7
130-139	19
120-129	37
110-119	79
100-109	69
90-99	65
80-89	17
70-79	5
60-69	3
50-59	2
40-49	1

Comments

Abhishek says

December 6, 2017 at 2:12 am

If class interval is varying then how to find the value of h

Jothikumar says

August 1, 2018 at 3:07 am

I am using of information

gayathri says

March 11, 2019 at 3:01 pm

Modal class of continuous variable is class associated with highest, lowest or which else?

Nishita says

July 1, 2019 at 2:13 pm

What if the last class interval has the highest frequency? Then what will be f2?

Rajaram Yadav says

July 18, 2020 at 6:04 am

f2=0

Rajni says

January 7, 2021 at 5:23 am

Find the mode from the following frequency distribution.

Number 8 9 10 11 12 13 14 15 16
Frequency 3 8 12 15 14 17 12 8 6

Kindly show me the answer in steps.