Mode Formula: Types, Derivation, Grouped & Ungrouped Data

Mode formula is one of the measures of central tendency to determine the value of a set of data. It tends to provide information about items that are present in the data set and about the ones that occur frequently in the set.

• In simple terms, the mode is the value that occurs most frequently in the data set.
• Mode is also known as modal value. A given set of data may have one mode or more than one mode.
• In statistics, mode plays an important role to identify which is the one value that occurs most frequently. 
• The mode of a set of data is the value that appears most frequently in the data set.

The formula for calculating the mode is:

The mode is particularly useful for datasets with discrete values, such as integers or categories, where the frequency of occurrence is of interest. To find the mode, one can simply identify the value that appears most frequently in the dataset.

Also Check: Difference Between Variance and Standard Deviation

Key Terms: Statistics, Dataset, Frequency, Cental Tendency, Grouped Data, Frequency Distribution, Modal Class, Ungrouped data, Measures of Dispersion, Ordered Set, Mean, Median

What is Mode?

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Mode, in statistics, is defined as the value that has a higher frequency in a given set of values. The value that appears the most number of times is mode or modal value.

• In statistics, we represent a data set by a representative value that roughly defines the entire data collection and this representative value is known as the measure of central tendency.
• Central tendency is the value around which the data is centred and it allows us to create a statistical summary of the organized data.
• One such measure of central tendency is called the mode of data. 

Mode of a Dataset

Mode Formula Derivation 

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We consider the modal class of size, determining the mode on basis of it. For the same, consider the graph as represented below. Let the modal class frequency be f_m or f₁. Here, as can be seen, BC = h.

Also, assume that the preceding modal class is f₀, while the frequency of the class succeeding the modal class is f₂. Besides, consider that the lower limit of the modal class is I₀. Hence, the mode can be represented by I₀ + x.

As per the diagram given, it can be seen, △AEB ~ △DEC

Hence, AB/CD = BE/DE = \(\frac{f_{1}-f_{0}}{f_{1}-f_{2}} \)

Also, △BEF ~ △BDC

⇒ FE/BC = BE/BD = \(\frac{f_{1}-f_{0}}{\left(f_{1}-f_{0}\right)+\left(f_{1}-f_{2}\right)}=\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}} \)

FE = \(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}} \times B C=\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h \)

Thus, it can be seen that:

x = \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h \)

Hence, the mode can be shown by:

I₀ + x = I₀ + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h \)

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Types of Mode

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There are different types of mode as a dataset may have more than mode. The resective types of mode are discussed below:

Unimodal Mode

A set of data with one mode is known as unimodal mode.

For Example: In data set A = { 14, 15, 16, 17, 15, 18, 15, 19} there is only one mode 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. It means that there are two data values that are having the highest frequencies.

For Example: In the data set A = { 8,12,12,14,15,17,17,19} there two modes,12 and 17 because both 12 and 17 are repeated 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. It means that there are three data values that are having the highest frequencies.

For Example: In the data set A = {2, 2, 2, 3, 4, 4, 5, 6, 5,4, 7, 5, 8} there are three modes, 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: In the data set A = {100, 80, 80, 95, 95, 100, 90, 90, 102, 95 } is 80, 90, 95, and 102 because all four values are repeated twice in the given set. Hence, it is a Multimodal data set.

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How to Find Mode Formula?

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To find the mode of a set of data, the value or values that occur most frequently should be identified first. Here are the steps to find the mode:

• Organize the data in numerical order.
• Count the number of times each value appears in the data.
• Identify the value or values that occur most frequently. These are the modes.

In case there is one value occurring most frequently, then the data has a single mode. If two or more values occur most frequently, then the data has multiple modes.

For example, assume the following set of data: 2, 3, 4, 4, 4, 6, 7, 7, 7, 9, 9, 7, 7

Then organize the data in numerical order: 2, 3, 4, 4, 4, 6, 7, 7, 7, 7, 7, 9, 9

Further, count the number of times every value occurs in the data:

• 2 appears once
• 3 appears once
• 4 appears thrice
• 6 appears once
• 7 appears five times
• 9 appears twice

The value 7 appears most frequently, which is five times, so the mode is 7.

Mode Formula for Ungrouped Data

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Mode for ungrouped data can be found easily after arranging the data values either in ascending or descending order, so that we can easily recognise the repeated values and their frequency. Hence, the observation with the highest frequency will be the mode of the given data. Alternatively, we can form a frequency distribution table to get the mode.

Thus, the mode formula for ungrouped data is:

To understand better an example is illustrated below.

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Mode for Grouped Data

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In a grouped frequency distribution, unlike ungrouped data, it isn’t possible to determine the mode by looking at the frequencies. In this case, we can only locate a class with the maximum frequency, called the modal class. The mode is a value that lies in the modal class and is calculated using the formula given below:

Here,

l = Lower limit of the modal class

h = Size of the class interval (assuming all class sizes to be equal)

f₁ = Frequency of the modal class

f₀ = Frequency of the class preceding the modal class

f₂ = Frequency of the class succeeding the modal class

Things to Remember

• Mode is one of the measures of central tendency to determine the value of a set of data.
• In statistics, we represent a set of data by a representative value that roughly defines the entire data collection and this representative value is known as the measure of central tendency.
• A set of data with one mode is known as unimodal mode, with two modes bimodal mode and so on. 
• A set of data with four or more than four modes is known as a multimodal mode.
• The mode of grouped data is given by the formula \( l+{{f_1-f_0} \over 2f_1 - f_0 - f_2}\times{h}\).

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