Measures of Central Tendency: Mean, Median, Mode

A single number representing a set of data by identifying the central position within that set is a measure of central tendency. As a result, central tendency measures are also known as central location measures.

Central tendency focuses on the central distribution of data through a single value. Types of central tendency in real-life are average marks, rainfall, income, etc. Three commonly used measures of central tendency such as arithmetic mean, median, and mode.

This is the most important statistical method to represent the value of the dataset. We can choose the measures of central tendency based on the kind of data we have. Students learn this concept in order to solve mathematical problems in a fraction of seconds. You can check NCERT Solution for Class 11 Maths Chapter 15 for better understanding. We have provided detailed information on measure of central tendency in this article. Read on to find out about its definition, types, and examples.

Measures of Central Tendency: Definition

The measure of central tendency is the summary of the data set in the form of a typical value. It focuses on providing an accurate description of the data. This is one of the important methods to solve statistical problems. The measures of central tendency provide a summary of the data set rather than the individual data.

Understanding Mean, Median And Mode

There are several measures of central tendency, out of which the most commonly used are Mean, Median and Mode. Let us now focus on the most commonly used measures of central tendency as mentioned below:

Mean

Mean is defined as the sum of values in the dataset divided by the number of observations or values. This is the most commonly used measure of central tendency and is also called the arithmetic mean. It is calculated by using the formula:

X1 + X2+ X3 +……………….+ XN/ N

There are two types of arithmetic mean as the arithmetic means of ungrouped data and arithmetic mean for grouped data.

For example, A monthly income of 4 families is 1600, 1400, 1300, 1200. Find the average income of the family.

Solution: Given, 1600 + 1400 + 1300 +1200

= 1600 + 1400 + 1300 +1200 /4

= 1375

Therefore, the average monthly income of the family is Rs. 1375.

Median

It is defined as the value that divides the distribution into two halves. One part of the value is greater or equal to the median value and the other is less than or equal to it. You can calculate the median by arranging the data from the smallest value to the largest one.

For example, Arrange the data 5, 7, 6, 1, 8, 10, 12, 4, and 3 in ascending order and find out its median.

1, 3, 4, 5, 6, 7, 8, 10, 12

The middle value of the data is 6. Here, half of the numbers are greater than 6 and the other halves are smaller.

In case, the middle value has two numbers in the data, you calculate by using the formula mentioned below:

1, 3, 4, 5, 6, 7, 8, 10, 12 , 13

This can be calculated by adding two median values divided by the number of observations. For example, 6+7/ 2= 9.5

Therefore, the median value is 9.5.

Mode

Mode is the frequently observed data value. There are chances that we get repeated numbers in the data set.

For example, 1, 2, 3, 4, 4, 5, Here 4, 4 is the frequently observed value.

Examples of Central Tendency

Some of the examples of measures of central tendency are given below:

Example 1: Find the mean deviation about the mean for the data 4,7,8,9,10,12,13,17

Solution: Step 1:The given data is 4,7,8,9,10,12,13,17

Mean of the data, ?̅= 4+7+8+9+10+12+13+17/ 8 = 80/ 8 = 10

The deviations of the respective observations from the mean ?̅, i.e. ?? − ?̅ are – 6, – 3, – 2, – 1,0,2,3,7

The absolute values of the deviations, i.e. |?? − ?̅|, are 6,3,2,1,0,2,3,7

The required mean deviation about the mean is

M.D. (?̅) = ∑ |?? − ?̅| / 10

12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6 / 10 = 84 /10 = 8.4

Example 2: Find the mean deviation about the mean for the data 4,7,8,9,10,12,13,17

Solution: Step 1:The given data is 4,7,8,9,10,12,13,17

Mean of the data, ?̅= 4+7+8+9+10+12+13+17/ 8 = 80 /8 = 10

The deviations of the respective observations from the mean ?̅, i.e. ?? − ?̅ are – 6, – 3, – 2, – 1,0,2,3,7

The absolute values of the deviations, i.e. |?? − ?̅|, are 6,3,2,1,0,2,3,7

The required mean deviation about the mean is

M.D. (?̅) = ∑ |?? − ?̅|/ 8

= 6 + 3 + 2 + 1 + 0 + 2 + 3 + 7/ 8 = 24 /8 = 3

FAQs

The frequently asked questions on measures of central tendency are given below:

Now we have provided information on measures of central tendency in this article. So, study seriously and master every concept in the CBSE Class 11 syllabus. Make use of these NCERT solutions for Class 11. You can also solve CBSE Class 11 PCM questions on Embibe. These will help you in your Class 11 preparation as well as other competitive exams.

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