Calculating Mode

A common measure of central tendency, the mode is the frequency at which a given number occurs in a series of values. Derived from the French word ‘La Mode’ (meaning: fashionable), it is also sometimes known as ‘modal value’. 

One of the most useful measures of central tendency, this comes in handy, especially when you’re dealing with categorical data. A dataset can have only a single mode, more than one mode, or even no mode at all.

Definition & Meaning of Mode

As stated above, the mode is the value that appears the maximum number of times in a series. Statistics is a field that deals with the collection, sorting, and analysis of both grouped and ungrouped data. 

This data is then presented in a format that can be easily understood by all. To that end, we often use a single value to represent an entire series of data, which is known as ‘mode’. 

Like all the other measures of central tendency, it too helps us create a detailed summary from the vast amount of large, complex data. 

Example: In a given set of values, say – {2,4, 6, 8, 6, 9, 10}; the mode for this series shall be 6. 

Why? Because in the above series, 6 is the only number that appears a maximum number of times. 

Types of Mode 

The mode can be classified into three types based upon its occurrence in a dataset. They are as follows: 

 1. Unimodal mode 

Derived from the Latin word ‘uni’ which means one. In simple words, a ‘unimodal mode’ is one that only has a single mode. 

For example, in a set A, the values are as follows – [12, 15, 12, 14, 9, 12, 8]. Here, there is only one number that is being repeated in the series. 

Hence, this series is considered as a ‘unimodal data set’. 

 2. Bimodal mode 

Originating from the Latin word ‘bis’, this is a set of data where there are two modes that will be considered to be a bimodal model. This means that two values have the highest frequency. 

For example, in a set A, the values are as follows – [12, 12, 12, 14,14,14, 9, 12, 8]. Here, there are  two number ( 12 and 14 ) that are being repeated 3-times in the series 

 3. Trimodal mode 

As evident from the name, a data set that has 3 modes with the highest frequencies is said to be a ‘trimodal mode’. 

For example, in a set A, the values are as follows – [12, 12, 12, 14,14,14, 9,9,9 , 12, 8]. Here, there are  three  numbers  (12 , 14 and 9) that are being repeated 3-times in the series 

 4. Multimodal mode

A data set with four or more modes will be known as ‘multimodal mode’. 

What is the formula for mode? 

While there’s no specific formula for calculating the mode of an ungrouped series, the mode for a grouped series is calculated using a formula that we will be discussing below.

Mode Formula in Statistics (Ungrouped Data)  

Ungrouped data is the raw data or observations you gather from an experiment or study. It is scattered and hasn’t been classified or groups as yet. As stated above, the number appearing most frequently in a series is its mode. 

A data set can have more than one frequency as well since it is a possibility that two observations have the same frequency. In such cases, the data is considered to be ‘multimodal’. 

This is better explained using an example: 

E.g.- The table given below represents the number of goals scored by a footballer in 8 matches. Find the mode of the given series: 

Solution: 

We can see that the footballer has scored 2 goals frequently in many matches he has played. Therefore, the mode for this series will be 2. 

Calculating Mode for Grouped Data 

Grouped data is usually given in the form of class intervals such as 0-10, 10-20, 20-30, and so on. It is formed by collecting the raw, ungrouped data into different groups so that the frequency distribution serves as a means of summarization. 

For grouped data, one needs to calculate the modal class to determine its mode, since it lies inside the given class interval. 

The formula for calculating mode is as follows: 

Mode = l +f1- f02f1-f0-f2h

Here,

l= lower limit of the class interval 

f0  = Frequency of class before the modal class 

f1 = Frequency of modal class 

f2= Frequency of class after the modal class 

h = Size of the class interval 

How to Calculate Mode in A Series? 

There are multiple ways to calculate the mode of a given series. In the next few paragraphs, we shall be covering the different types of series, along with the formula for calculating the modes of each. 

  1. Individual Series 

An individual series is one where all the numbers/values are listed individually, in a raw and unorganized manner. Raw data is then arranged in descending or ascending order to obtain a certain result. 

E.g. – 8,2,4,5,6,2,4,3,4,7,8,4,6,1,4,6,2

When arranged in ascending order, it gives the following answer: 1,2,2,2,3,4,4,4,4,4,5,6,6,6,7,8,8. 

  2. Discrete Series

If the values are repeated consistently, it is important to organize them in a tabular form to represent the values and the number of times they have been repeated, which is also known as their ‘frequency’. 

To calculate mode for a discrete series, you just need to identify the variables with the highest frequency incurred.  

E.g. – Take a look at the following discrete series. 

The arithmetic mode of the aforementioned series is 80 since the highest frequency (12) is associated with it. 

  3. Continuous Series 

Also known as frequency distribution, here frequencies are given alongside the value of a variable in the form of class intervals. 

The formula for calculating the mode of a continuous series is the same as that for grouped data. 

Conclusion 

So, this was all about the most common measure of central tendency used in statistics (mode). We not only covered the meaning of mode but also dived deep into how you can calculate the mode for different types of grouped and ungrouped data.