Mode of Grouped Data: Formulas and Solved Examples

In statistics, the mode (aka modal value) is one of the three measures of central tendency. It is defined as the value that occurs repeatedly in a given data set. A set can have one, two, or more than two modes, or even no mode at all.

Mode is the number or value that appears most frequently or the highest number of times in a data set.

For example, the modal value of the set {2, 3, 3, 7, 5, 8, 4, 3, 9} is 3.

Keyterms: Mode, central tendency, data set, mean, median, unimodal, bimodal, trimodal, and multimodal mode

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Definition of Mode

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In statistics, we represent the data using tables, graphs, pie charts, etc. To analyze the represented data and understand it further, we define the entire collection of data roughly using a representative value. This representative value is called the measure of central tendency.

There are three main measures of central tendency. They are mean, median, and mode.

Mode is the value that occurs most frequently in an observation. In simple terms, it is the value that appears the most number of times in a data set.

For the data set {4, 5, 6, 5, 2, 3, 7, 9} the mode is 5, as it appears twice in the set.

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Types of Mode

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There are different types of modes like unimodal, bimodal, trimodal, and multimodal mode.

Unimodal Mode

A data set with one mode is called a unimodal mode.

For example, Consider the set A = {3, 4, 5, 3, 1, 7, 3, 6}.

Here, only the value 3 appears three times in the given set. Hence, the set has only one mode value which is 3. So it is a unimodal set.

Bimodal Mode

A data set with two mode values is called a bimodal mode. In such a data set, there are two values with the highest frequency.

For example, Consider the set B = {3, 4, 5, 3, 1, 7, 5, 2, 8, 5, 3, 6}.

Here, the values 3 and 5 both are repeating three times. Hence, the set has two modal values 3 and 5. So, it is a bimodal set.

Trimodal Mode

A data set with three modes is called a trimodal mode. In such a data set, there are three values with the highest frequency.

For example, Consider the set C = {4, 6, 7, 4, 3, 4, 5, 3, 1, 7, 5, 2, 8, 5, 3, 6}.

Here, the values 3, 4, and 5 are repeating three times. Hence, the set has three modal values 3, 4, and 5. So, it is a trimodal set.

Multimodal Mode

A data set with four or more modes is called a multimodal mode. In such a data set, there are four or more values with the highest frequency.

For example, Consider the set D = {7, 8, 2, 7, 4, 6, 7, 4, 3, 4, 5, 3, 1, 7, 5, 2, 8, 5, 3, 6}.

Here, the values 3, 4, 5, and 7 are repeating three times. Hence, the set has four mode values 3, 4, 5, and 7. So, it is a multimodal set.

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Formula of Mode

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We cannot find the mode of a continuous series or grouped data, by looking at the data set. For that, we need to calculate the modal class, as the mode lies in the modal class. The formula for calculating the mode for a grouped data is as follows:

Mode = 

\(I = \frac{f_1 - f_0}{2f_1 - f_2 - f_0} \times h\)

Where,

l = lower level of the modal class

h = size of the class interval

f₁= frequency of the modal class interval

f₀= frequency of the class interval preceding the modal class

f₂ = frequency of the class interval succeeding the modal class

The steps to calculate the mode of grouped data with equal class intervals using the above formula is as follows:

Step 1: Prepare the frequency distribution table such that the observations are in the first column and their respective frequency is in the second column.

Step 2: Find out the class that has the maximum frequency. This class is your modal class.

Step 3: Find the class size by computing: upper limit - lower limit.

Step 4: Calculate the mode using the above formula.