Mode Formula
What is Mode?
The mode is the observation’s value, which occurs most frequently, i.e., an observation with the maximum frequency is called the mode. A data set can have more than one mode, which means more than one observation has the same maximum frequency. In this article, mode formulas for grouped and ungrouped data are explained with the solved examples.
Learn what is the mode in detail here.
Mode Formula For Ungrouped Data
To find the mode for ungrouped data, it would be better to arrange the data values either in ascending or descending order, so that we can easily find 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:
The most frequently occurred value in the data set
Also, read: |
Mode Formula for Grouped Data
Now, let us discuss the way of obtaining the mode of grouped data. As we know, 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 will solve problems having only a single mode.
In a grouped frequency distribution, unlike ungrouped data, it is impossible 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 that lies in the modal class and is calculated using the formula given as:
\(\large Mode = l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\)This is the mode formula for grouped data in statistics.
Here,
l = Lower limit of the modal class
h = Size of the class interval (assuming all class sizes to be equal)
f_{1} = Frequency of the modal class
f_{0} = Frequency of the class preceding the modal class
f_{2} = Frequency of the class succeeding the modal class
Mode Formula Class 10
In Class 10 maths, the mode formula is given for grouped data. However, the formula is suitable for the data having a single mode. Several solved examples and practice problems have been provided in Chapter 14 of the curriculum. All these problems will help to improve the knowledge of one of the measures of central tendency, i.e. mode.
Solved Examples
Go through the examples provided below for a better understanding of the concept and formulas explained above.
Example 1: Find the mode of the following marks obtained by 25 students in a mathematics test out of 50.
34, 46, 45, 39, 43, 22, 27, 37, 46, 35, 34, 39, 40, 30, 30, 41, 37, 46, 39, 29, 34, 39, 35, 43, 30
Solution:
The ascending order of the data:
22, 27, 29, 30, 30, 30, 34, 34, 34, 35, 35, 37, 37, 39, 39, 39, 39, 40, 41, 43, 43, 45, 46, 46, 46
The most frequently occurred value is 39.
Hence, the mode of given marks is 39.
Alternatively, let us form the table with observations and their frequencies to get the mode.
The mode of the given data can be obtained by making the frequency table and choosing the highest frequency. Such as:
Observation | 22 | 27 | 29 | 30 | 34 | 35 | 37 | 39 | 40 | 41 | 43 | 45 | 46 |
Frequency | 1 | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 3 |
Here, the highest frequency is 4.
Therefore, the mode is 39.
Example 2: Calculate the mode of the following frequency distribution.
Class | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
Frequency | 7 | 14 | 13 | 12 | 20 | 11 | 15 | 8 |
Solution:
From the given table,
The highest frequency = 20
This value lies in the interval 50-60. Thus, it is the modal class.
Modal class = 50 – 60
l = Lower limit of the modal class = 50
h = Size of the class interval (assuming all class sizes to be equal) = 10
f_{1} = Frequency of the modal class = 20
f_{0} = Frequency of the class preceding the modal class = 12
f_{2} = Frequency of the class succeeding the modal class = 11
Mode \(=l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\) \(=50+\frac{20 – 12}{2\times 20 – 12 – 11}\times 10\)
= 50 + [80/ (40 – 23)]
= 50 + (80/17)
= 50 + 4.706
= 54.706
Therefore, the mode is 54.706.
Practice Problems
- Calculate the mode for the data: 8, 2, 3, 5, 4, 2, 8, 2, 5, 3, 8, 5, 6, 3, 2, 3, 8, 5, 5, 6
- Find the mode of the following distribution:
CI 45 – 55 55 – 65 65 – 75 75 – 85 85 – 95 Frequency 3 10 11 8 3 - What is the mode of the following distribution?
Class 0 – 5 6 – 11 12 – 17 18 – 23 24 – 29 Frequency 13 10 15 8 11
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