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Introduction to the Measures of Central Tendency (Mean, median and mode)

Let us consider the following example.

Mohit and Rohit are the opening batsmen for their school cricket team. The following table shows the runs scored by them in the last 10 innings.

Mohit

74

5

55

48

99

105

30

17

33

54

Rohit

42

101

51

38

53

100

105

44

72

41

Can you say who is a better batsman by observing the table?

From the given table, we observe that both of them scored a maximum of 105 runs in a match. However, this does not tell us anything.

Now, we can see that the highest runs scored by Mohit are 105, while the lowest runs scored by him are 5.

Therefore, the difference between the highest and the lowest runs scored by Mohit is

105 − 5 = 100 runs.

Hence, the range of runs scored by Mohit is 100 runs.

The difference between the highest and the lowest values of a data set is called the range of the data set.

Similarly, the range of the runs scored by Rohit is

Highest score − Lowest score

= 105 − 38

= 67 runs

Thus, we can see that while their maximum scores were equal, the range of Rohit’s scores was lesser than the range of Mohit’s scores. What does this tell us? This tells us that Rohit was more consistent than Mohit, scoring a minimum of 38 runs in each match.

Now, let us discuss some more examples based on the above concept.

Example 1:

Find the range of the following data:

210, 150, 162, 190, 26, 175, 200, 216, 50, 127, 116, 100

Solution:

Here, highest value = 216

Lowest value = 26

The range of the given data = 216 − 26

= 190

Example 2:

The weekly temperature of Delhi is shown below.

Days

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Temperature (in °Celsius)

49

45

40

35

42

46

48

What is the range of temperatures?

Solution:

Here, the highest temperature is 49°C and the lowest temperature is 35°C.

Therefore, range of temperatures = Highest temperature − Lowest temperature

= 49°C − 35°C

= 14°C

Let us consider the following example.

Mohit and Rohit are the opening batsmen for their school cricket team. The following table shows the runs scored by them in the last 10 innings.

Mohit

74

5

55

48

99

105

30

17

33

54

Rohit

42

101

51

38

53

100

105

44

72

41

Can you say who is a better batsman by observing the table?

From the given table, we observe that both of them scored a maximum of 105 runs in a match. However, this does not tell us anything.

Now, we can see that the highest runs scored by Mohit are 105, while the lowest runs scored by him are 5.

Therefore, the difference between the highest and the lowest runs scored by Mohit is

105 − 5 = 100 runs.

Hence, the range of runs scored by Mohit is 100 runs.

The difference between the highest and the lowest values of a data set is called the range of the data set.

Similarly, the range of the runs scored by Rohit is

Highest score − Lowest score

= 105 − 38

= 67 runs

Thus, we can see that while their maximum scores were equal, the range of Rohit’s scores was lesser than the range of Mohit’s scores. What does this tell us? This tells us that Rohit was more consistent than Mohit, scoring a minimum of 38 runs in each match.

Now, let us discuss some more examples based on the above concept.

Example 1:

Find the range of the following data:

210, 150, 162, 190, 26, 175, 200, 216, 50, 127, 116, 100

Solution:

Here, highest value = 216

Lowest value = 26

The range of the given data = 216 − 26

= 190

Example 2:

The weekly temperature of Delhi is shown below.

Days

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Temperature (in °Celsius)

49

45

40

35

42

46

48

What is the range of temperatures?

Solution:

Here, the highest temperature is 49°C and the lowest temperature is 35°C.

Therefore, range of temperatures = Highest temperature − Lowest temperature

= 49°C − 35°C

= 14°C

Bhangra’s is a very popular shop that sells watches of foreign brands in Delhi’s posh Connaught Place market. The owner of the shop decided to stock the brand whose watches were selling the most. He decided to look at the previous month’s sales, which is listed as

Brand

Number of watches sold

Alpha

17

Townzen

23

Tag Heuim

7

Twatch

13

Based on this information, the owner decided to stop keeping Tag Heuim watches because very few people buy them. He also decided to keep more varieties of Townzen watches because most people were buying this brand.

In this data set, the highest occurring event (23) corresponds to Townzen watches and is known as the mode of this data set. Just like mean and range, mode is another measure of central tendency of a group of data. It can be defined as

The value of a set of data that occurs most often is called the mode of the data.

Let us find the mode of another data set in the given video.

Here, the data set did not contain too many terms and could thus be easily arranged in ascending order. However, in case of very large data, it is not always easy to arrange it in ascending or descending order. Therefore, in such cases, it is better to arrange the data set in the form of a table with tally marks.

So let us understand the usage of tally marks method to find the mode with the help of the given video.

A collection of data can have more than one mode. The data sets having one mode or two modes or more than 2 modes are  said to have uni-mode or bi-mode or multi-mode. 

Let us now look at some more examples to understand this concept better.

Example 1:

Find the mode of the following numbers.

2, 6, 7, 5, 4, 2, 6, 7, 9, 7, 8, 3, 2, 11

Solution:

The given set of numbers can be arranged in ascending order as

2, 2, 2, 3, 4, 5, 6, 6, 7, 7, 7, 8, 9, 11

Here, 7 and 2 occur most often (3 times). Therefore, both 7 and 2 are the modes of the given set of numbers.

Example 2:

Find the mode of the following data set.

1000, 200, 700, 500, 600, 160, 270, 300, 360, 950

Solution:

The increasing order of the given numbers is

160, 200, 270, 300, 360, 500, 600, 700, 950, 1000

Here, every number is occurring only once.

Thus, the given data has no mode.

Note: The above example shows that the mode of a data may or may not be unique. Also, there are some data sets which do not have any mode.

Example 3:

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