# What is the formula of mode?

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## Mode: Definition, Formulas, Types and Solved Examples

Shashank Ranguwar | Updated: Dec 21, 2021 12:26 IST

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We come across data everywhere; in the newspapers, articles, in our usual bank statements, mobile and electricity bills and more. The mean, median and mode are termed as the three measures of central tendency. Mean is defined as the arithmetic average of a data set. Mean is determined by adding the numbers/values in a data set and dividing the total by the number of observations in the data set.

The median is the middle character in a data set when the numbers are presented in ascending or descending order. Whereas the mode is the value that happens to appear most often in a data set and the range is the difference between the highest and lowest values in a data set.

With this article on mode, we will aim to understand what is the mode? or mode definition along with the mode formula, how to find the mode and so on.

If you are reading Mode, you can also read about the Three Dimensional Geometry here.

### Mode: Meaning

In Statistics, mode or modal value is that observation that happens at the maximum time or has the highest frequency in the provided set of data. A set of numbers/data with one mode is known as unimodal, a set of numbers possessing two modes is bimodal and a set of numbers having three modes is known as trimodal. However, any set of numbers holding four or more than four modes is known as multimodal.

## Mode Formula

Before moving towards the various mode formulas in statistics let us understand the different types of modes: namely unimodal, bimodal, trimodal, and multimodal.

**Unimodal Mode** A collection of data/numbers with one mode is recognised as a unimodal mode.

For example, the mode of data set B = { 20, 14, 16, 17, 14, 18, 14, 19} is 14 as there is simply one value replicating itself. Therefore, it is a unimodal data set.

**Bimodal Mode** A set of data including two modes is identified as a bimodal model. This indicates that there are two data values that possess the highest frequencies.

For example, the mode of data set B = { 8, 12, 12, 14, 15, 19, 17, 19} is 12 and 19 as both 12 and 19 are repeated twice in the given set. Therefore, the given set is a bimodal data set.

**Trimodal Mode ** A collection of data including three modes is identified as a trimodal mode. This implies that there are three data values that are holding the highest frequencies.

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

**Multimodal Mode** A set of data including four or more than four modes is recognised as a multimodal model.

For example, The mode of data set B = {101, 82, 82, 95, 95, 100, 90, 90,101 ,96 } is 82, 90, 95, and 101 because all the four values are recurring twice in the given set.

Also, learn the concepts of lines in detail here.

Now moving towards the mode formula; for **ungrouped data**, we only need to identify the observation which occurs at maximum times.

**Mode = Observation with maximum frequency**

For example in the data set: 7, 8, 9, 2, 4, 7, 7, 6, 3 the value 7 appears the most number of times.

Thus, the mode is equal to 6 for the set.

**Key point**: A data may possess no mode, one mode, or more than one mode depending upon the data set.

For grouped data or when the data is continuous, the mode can be determined using the following rules:

**Step 1: **Find the modal class i.e. the class with the highest frequency.

**Step 2:** Find mode applying the following formula:

Where,

Also, learn about Sequences and Series here.

## How to Find the Mode?

Mode is the value that happens most commonly in a given set of data. Let us understand how to find the mode for a given set of data with examples:

**Example 1**: Determine the modal value for the given set of data.1214201822162020111216142022

Solution:

Arranging the above set of data in ascending order.1112121414161618202020202222

Here, we get 20 four times, 12, 14, 16, 22 twice each, and other terms i.e 11 and 18 only once.

Therefore, the mode for a given set of data is 20.

**Example 2: **Determine the mode value for the given data.

**Example 3**: What is the mode value of the given scores?

8, 8, 4, 3, 4, 8, 5, 5

Solution:

- Mode: The word mode has been derived from the French word la Mode which signifies the most fashionable values of distribution because it is repeated the highest number of times in the series. The mode is the most frequently observed data value. It is denoted by Mo.
- The mode is seldom used and its computation is easy, but it is highly unstable and may change with minor shifts in the frequencies from one interval to another.
- However, there are situations in which the only mode can be used.
- For example, if a shoe company wants to show which size of shoe it should produce more, it would use mode as a measure of central tendency. The most frequently sold size of the shoes is the mode.
- In the above situation, we can see that 8 is repeated the highest number of times, so the mode is 8.

Hence, we conclude that in the above situation the mode is 8.

**Example 4**: The mode of the following data is 8, 12, 6, 13, 7, 6, 8, 12, 9.

**Solution:**

Given data is 8, 12, 6, 13, 7, 6, 8, 12, 9

Concept:

The mode is the value that appears most often in a set of data.

Calculation:

Here, we see that for the given data 8, 12, and 6, these three values come two times.

Hence, we conclude that in the above situation the modes are 8, 12, and 6. This states that the above data is an example of the trimodal class.

A collection of data including three modes is identified as a trimodal mode. This implies that there are three data values that are holding the highest frequencies.

## Relationship Between Mean, Median and Mode

The three measures of central values i.e. mean, median, and mode are closely correlated by the following relations, also called an **empirical relationship**.

**2 Mean + Mode = 3 Median**

or

**Mode = 3 Median 2 Mean**

For example, if we have data whose mode =60 and median = 61. Then, we can determine the mean using the above relation.

2 Mean + Mode= 3 Median

2Mean + 60= 3 x 61

Mean = 123/2 = 61.5

Mean = 61.5

Check out this article on the Binomial Theorem.

## Key Takeaways from Mode Concept

- In statistics, the mode is the value that is most commonly observed in a set of data.
- In many instances, the modal value will vary from the average value in the data.
- The mode is simple to understand and determine.
- The mode is not modified by extreme values.
- The mode is beneficial for qualitative data.
- The mode can be positioned graphically.
- The mode is not determined when there is no repetition in a data set.
- Sometimes data sequences may hold one mode, more than one mode, or no mode at all.

We hope that the above article on Mode is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.**If you are checking Mode article, also check the related maths articles in the table below:**MagnitudeTrigonometric functionsHarmonic meanRolles theoremIdentity matrixSubsets

## Mode FAQs**Q.1What is the mode in math?**

**Ans.1 **The number that appears for the maximum time in a set of numbers is called the mode for that particular set. Example: in {6, 2, 9, 6, 6, 4, 9, 2} the mode is 6 as it has occurred for the maximum time.**Q.2What are the types of modes?**

**Ans.2 **The different types of modes are unimodal, bimodal, trimodal, and multimodal depending upon the data set provided.**Q.3Can the Number zero be a Mode?Ans.3 **Yes, the number zero can be a mode if it appears more than once in a given set of data.**Q.4What is the relation between mean, median, and mode?**

**Ans.4 **Relationship between Mean, Median, and mode:

Mode = 3 Median 2 Mean**Q.5How do I calculate the mean?**

**Ans.5 **The mean is the average of the numbers. To calculate the mean; add up all the numbers/data, then divide by the total numbers present. In other words, it is the sum divided by the total count.**Q.6What is the meaning of mode?**

**Ans.6 **The mode is the value that arises most frequently in a data set. A set of data may possess one mode, more than one mode, or no mode at all.**Q.7What is the formula for mean calculation ?Ans.7 **The formula of mean is expressed as Mean = {Sum of Observation} ÷ {Total number of Observations}

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