Mode of Grouped Data

Recall from Class IX, a mode is that value among the observations which occurs most often, that is, the value of the observation having the maximum frequency. Further, we discussed finding the mode of ungrouped data. Here, we shall discuss ways of obtaining a mode of grouped data. It is possible that more than one value may have the same maximum frequency. In such situations, the data is said to be multimodal. Though grouped data can also be multimodal, we shall restrict ourselves to problems having a single mode only.

Example 4

The wickets taken by a bowler in 10 cricket matches are as follows:

Solution

Let us form the frequency distribution table of the given data as follows:

Clearly, 2 is the number of wickets taken by the bowler in the maximum number (i.e., 3) of matches. So, the mode of this data is 23.

In a grouped frequency distribution, it is not possible to determine the mode by looking at the frequencies. Here, we can only locate a class with the maximum frequency, called the modal class. The mode is a value inside the modal class, and is given by the formula:

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Mode = l + f₁ - f₀2f₁-f₀-f₂ x h

where 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.

Let us consider the following examples to illustrate the use of this formula

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Example 5

A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household:

Here the maximum class frequency is 87,

and the class corresponding to this frequency is 3 – 5. So, the modal class is 3 – 5.

Now modal class = 3 – 5, lower limit (l) of modal class = 3, class size (h) = 2

frequency (f₁) of the modal class = 87,

frequency (f₀) of class preceding the modal class = 75,

frequency (f₂) of class succeeding the modal class = 23.

Now, let us substitute these values in the formula :

Mode = Mode = l + f₁ - f₀2f₁-f₀-f₂ x h

3 + 8 - 72 x 8 - 7 - 2 x 2 = 3 + 27 = 3.2863.3

Therefore, the mode of the data above is 3.286.

Example6

Example 6

The marks distribution of 30 students in a mathematics examination are given in Table 13.3 of Example 1. Find the mode of this data. Also compare and interpret the mode.

Solution

Refer to Table of Example 1. Since the maximum number of students (i.e., 7) have got marks in the interval 40 - 55, the modal class is 40 - 55. Therefore,

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