Measures of Central Tendency, Mean, Median and Mode: Introduction

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Introduction

Measures of Central Tendency

Measures of Central Tendency

Statistics deals with the collection of data and information for a particular purpose. The tabulation of each run for each ball in cricket gives the statistics of the game. The representation of any such collection of data can be done in multiple ways, like through tables, graphs, pie-charts, bar graphs, pictorial representation etc.

Now consider a 50 over match going between 2 countries. One of the countries scored 370 runs by the end of first innings. We need to find the overall run rate which is good for such a score.

Measures of Central Tendency

Often in statistics, we tend to represent a set of data by a representative value which would approximately define the entire collection. This representative value is called as the measures of central tendency. The name itself suggests that it is a value around which the data is centered.

The measures of central tendency are given by various parameters but the most commonly used are mean, median and mode. These parameters are discussed below.

Mean, Median and Mode

Mean

Mean is the most commonly used measures of central tendency. It actually represents the average of the given collection of data. It is applicable for both continuous and discrete data.

It is equal to the sum of all the values in the collection of data divided by the total number of values.

Suppose we have n values in a set of data namely as Then the mean of data is given by:

It can also be denoted as:

Median

Median represents the mid value of the given set of data when arranged in a particular order.

Median: Given that the data collection is arranged in ascending or descending order, the following method is applied:

  • i) If number of values or observations in the given data is odd, then the median is given by observation.

  • ii) If in the given data set the number of values or observations is even then the median is given by the average of observation.

Mode

The most frequent number occurring in the data set is known as the mode.

Consider the following data set which represents the marks obtained by different students in a subject.

Consider the following data set which represents the marks obtained by different students in a subject

Name

Anmol

Kushagra

Garima

Ashwini

Geetika

Shakshi

Marks obtained (out of 100)

73

80

73

70

73

65

The maximum frequency observation is (as three students scored marks), so the mode of the given data collection is .

Let us see the difference between the mean median and mode through an example.

Example: The given table shows the scores obtained by different players in a match. What is mean, median and mode of the given data?

The given table shows the scores obtained by different players in a match. What is mean, median and mode of the given data?

Sr. No

Name

Runs Scored

1

Sachin

80

2

Yuvraj

52

3

Virat

40

4

Sehwag

52

5

Rohit

70

6

Harbhajan

1

7

Dhoni

6

Solution:

i) The mean is given by

Here given data

Number of data

We put the data on formula,

Sum of the data,

Divide 301 by 7,

The mean of the given data is 43.

ii) To find out the median let us first arrange the given data in ascending order

To find out the median let us first arrange the given data in ascending order

Name

Harbhajan

Dhoni

Virat

Yuvraj

Sehwag

Rohit

Sachin

Runs

1

6

40

52

52

70

80

As the number of items in the data is odd in number the median is observation.

⇒ Median observation

⇒ Median observation

iii) Mode is most frequent data which is

Example: 1

Find the median of the set of numbers: 1, 2, 3, 4, 5,6, 7, 8, 9 and 10.

Solution:

Here given the data,

We have to find the median,

So first of all, we arrange the data into ascending order but here all data already arrange in ascending order.

Formula for the find of median

Here

Here given data is in even number so we take the two middle numbers like

Example 2:

The following represents age distribution of students in an elementary class. Find the mode of the values: .

Solution:

We have to find the mode of given data

Mode means frequently repeated data,

Here in given data 9 is most repeating data.

Example 3: Find the mean of these set of numbers: 100, 1050, 320, 600 and 150.

Solution:

Here given data,

Formula for the mean

Number of data

Put the values,