## Mode

In Statistics, mode or modal value is that observation which occurs at the maximum time or has the highest frequency in the given set of data. The mode is derived from the French word La Mode which means fashionable. A given set of data may have one or more than one mode. A set of numbers with one mode is unimodal, a set of numbers having two modes is bimodal, a set of numbers having three modes is trimodal, and any set of numbers having four or more than four modes is known as multimodal.

For example, The following table represents the number of runs made by the batsman in the first 10 balls. Find the mode of the given set of data.

It can be seen that 2 runs are made by the batsman frequently in different balls. Hence, 2 is the mode of the given data set.

### Define Mode

Mode is defined as a value that occurs most frequently in a dataset.

For Example,

In {6, 9, 3, 6, 6, 5, 2, 3}, the mode is 6 as it occurs most often.

Similar to the statistical mean and median, mode is a way of representing important information about random variables or populations in a single number. In a normal distribution, the value of mode or modal value is the same as the mean and median whereas the value of mode in a highly skewed distribution may be very different.

Mode is the most useful measure of central tendency while observing the categorical data such as the most preferred flavours of soda or models of bikes for which average median values based on order cannot be calculated.

### Types of Mode

The different types of mode are unimodal, bimodal, trimodal, and multimodal. Let us understand each of these modes.

Unimodal Mode - A set of data with one mode is known as a unimodal mode.

For example, the mode of data set A = { 14, 15, 16, 17, 15, 18, 15, 19} is 15 as there is only one value repeating itself. Hence, it is a unimodal data set.

Bimodal Mode - A set of data with two modes is known as a bimodal mode. This means that there are two data values that are having the highest frequencies.

For example, the mode of data set A = { 8,13,13,14,15,17,17,19} is 13 and 17 because both 13 and 17 are repeating twice in the given set. Hence, it is a bimodal data set.

Trimodal Mode - A set of data with three modes is known as a trimodal mode. This means that there are three data values that are having the highest frequencies.

For example, the mode of data set A = {2, 2, 2, 3, 4, 4, 5, 6, 5,4, 7, 5, 8} is 2, 4, and 5 because all the three values are repeating thrice in the given set. Hence, it is a trimodal data set.

Multimodal Mode - A set of data with four or more than four modes is known as a multimodal mode.

For example, The mode of data set A = {100, 80, 80, 95, 95, 100, 90, 90,100 ,95 } is 80, 90, 95, and 100 because both all the four values are repeating twice in the given set. Hence, it is a multimodal data set.

### How to Find the Mode?

Mode is the value that occurs most frequently in a given set of data.

Let us understand how to find the mode for individual series, discrete series, and continuous series.

Following are the formulas for finding the mode or modal value for different series.

Individual Series: Simply observe the maximum number of times an individual observation appears.

Let us understand how to find the mode of individual series with an example:

Calculate the modal value for the following set of data.

Solution:

Arranging the given set of data in ascending order.

Here, we get 16 four times, 12 and 20 twice each, and other terms only once.

Hence, the mode for a given set of data is 16.

Discrete Series: Simply find the variables with the highest frequency incurred.

Let us understand how to find the mode of discrete series with an example:

A shoe company manufactured winter boots with the size as mentioned below in the frequency distribution:

Solution:

In the given frequency distribution, we can clearly see 42 has the highest frequency.

Hence, the mode for the size of winter boots is 42.

Continuous Series: The formula for finding the mode or modal value in continuous series is given below:

\[l = \frac{f_{1} - f_{0}}{2f_{1} - f_{2} - f_{0}} \times h\]

Where,

l is the lower level of the modal class.

f_{1} is the frequency of the modal class.

f_{2} is the frequency of class interval succeeding the modal class.

f_{0} is the frequency of class interval preceding the modal class.

h is the width of the class interval.

Let’s understand with an example:

Modal Class = 10 - 15 (This is the class with the highest frequency).

Lower level of the modal class (l) = 10

Frequency of the modal class (f

_{1}) = 7Frequency of class interval succeeding the modal class (f

_{2}) = 2Frequency of class interval preceding the modal class (f

_{0}) = 3Width of the class interval (h ) = 5

Finding the Mode

\[l = \frac{f_{1} - f_{0}}{2f_{1} - f_{2} - f_{0}} \times h\]

\[\text{Mode }= 10 + \frac{(7 - 3)}{2(7) - 2 - 3} \times 5\]

\[= 10 + \frac{4}{9} \times 5\]

\[= 10 + \frac{20}{9}\]

\[= 10 + 2.22\]

\[= 12.22\]

Hence, the modal value for the given frequency distribution is 12.22

### Solved Examples:

1. Calculate the modal value for the following set of data.

Solutions:

Arranging the data in ascending order, we get

As there are two repeating values, it is a bi-modal data.

Hence, the modal values for a given set of data are 86 and 88.

2. Find the mode for the frequency distribution given below:

Solution:

Modal Class = 30 - 40 ( This is the class with the highest frequency).

Lower level of the modal class (l) = 30

Frequency of the modal class (f

_{1}) = 55Frequency of class interval succeeding the modal class (f

_{2}) = 44Frequency of class interval preceding the modal class (f

_{0}) = 30Width of the class interval (h ) = 10.

**Finding the Mode**

\[l = \frac{f_{1} - f_{0}}{2f_{1} - f_{2} - f_{0}} \times h\]

\[\text{Mode }= 30 + \frac{(55 - 30)}{2(55) - 44 - 30} \times 10\]

\[= 30 + \frac{25}{36} \times 10\]

\[= 30 + \frac{250}{36}\]

\[= 10 + 6.944\]

\[= 36.944\]

Hence, the modal value for the given frequency distribution is 36.944.

Q1. What are the Advantages of Mode in Mathematics?

Ans. Some of the advantages of mode in Mathematics are discussed below:

It is simple to understand and easy to calculate.

It can easily be calculated for the open end frequency distribution.

It is a commonly used technique for calculating average values such as average marks of students, average wages of labours, etc.

It can also be found graphically.

It can never be affected by extreme values. Hence, it is a good representation of data.

Q2. What are the Disadvantages of Mode?

Ans. Some of the disadvantages of mode are as follows:

It cannot be clearly defined in the case of the multimodal series.

The value of mode is most affected by the fluctuation in sampling in comparison to the mean and median

Mode is not suitable for statistical analysis and algebraic calculations.

Modal value cannot be used to find the total of the whole series as in case of mean.

Mode is often determined as ill-defined, ill-definite, and indeterminate.

Q3. Can the Number 0 be a Mode?

Ans. Yes, the number 0 can be a mode if it occurs more than once in a given set of data.