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Statistics (Measures of Dispersion)

Solve problems related to ranges of data sets

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

 

Mean deviation about the mean () of an ungrouped data x1, x2, x3 … xn is given by the formula , where is the mean of x1, x2, x3 …. xn. Mean deviation about the median (M) of an ungrouped data x1, x2, x3 …. xn is given by the formula, where M is the median of x1, x2, x3 …. xn .

To understand these concepts better, let us go through the following video.

 

Solved Examples

Example 1:

Find the mean deviation about the median of the following data, if its mean deviation about the mean is .

9, 17, 13, 19, 26, 31, 53, 42, x, 18, 21, 23

Where, the mean of this data lies between 23 and 26, and x lies between 27 and 31.

Solution:

To find the mean deviation about the median of the given data, first of all, we have to find the value of x.

Now, the mean of this data is given by

Since mean () of the given data lies between 23 and 26,

For each xi less than or equal to 23, For each xi greater than or equal to 26,

Hence, the mean deviation about the mean is given by

It is given that

Hence, the given data is

9, 17, 13, 19, 26, 31, 53, 42, 28, 18, 21, 23

Arranging this data in the increasing order, we get

9, 13, 17, 18, 19, 21, 23, 26, 28, 31, 42, 53

The median of this data is given by

So, the mean deviation about the median is given by

Example 2:

For what value of a natural number n are the values corresponding to 25 times the mean deviation about the median of the first (2n − 1) even natural numbers and 24 times the mean deviation about the mean of the first 2n odd natural numbers equal?

Solution:

The first (2n − 1) even natural numbers are

2, 4, 6, 8 …… 2(2n − 1)

Since n is a natural number, (2n − 1) is an odd number.

The median (M) of the data 2, 4, 6, 8 …… 2(2n − 1) is given by

Hence, the mean deviation about the median of the first (2n − 1) even natural numbers is given by

The first 2n odd natural numbers are

1, 3, 5 ….. 4n …

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