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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

Quartile Deviation The range and quartile deviation are measures of dispersion that depend on the values of the variables at a particular position in the distribution. The range is based on extreme values in the distribution. It does not consider the deviation among the values. In order to study the variation among the values, the measure of inter-quartile range is used. Inter-Quartile Range = Third Quartile − First Quartile = Q3 − Q1 Quartile deviation is the half of the difference between third quartile, Q3 and first quartile, Q1 of the series. ∴ Quartile deviation = Q3-Q12 Quartile deviation gives half of the range of middle 50% observations. Quartile deviation is also known as semi-inter-quartile range. Calculation of quartile deviation 1. For an individual series, the first and third quartiles can be calculated using the following formula: Q1 = Value of n+14th ordered observation Q3 = Value of 3n+14th ordered observation 2. For a discrete series, the first and third quartiles can be calculated using the following formula: If N = ∑f, then Q1 = Value of N+14th ordered observation Q3 = Value of 3N+14th ordered observation 3. For a continuous series, the first and third quartiles can be calculated using the following formula: Q1=L+N4-c.f.f×h Q3=L+3N4-c.f.f×h Here, L = lower limit of the quartile class f = frequency of the quartile class h = class interval of quartile class c.f. = total of all the frequencies below the quartile class N = total frequency, ∑f Solved Examples Example 1: Find the quartile deviation for the following data. 15, 65, 30, 70, 50, 25, 40, 75, 45, 60 Solution: First, arrange the observations in ascending order, as shown below: 15, 25, 30, 40, 45, 50, 60, 65, 70, 75 Here, number of observations, n = 10 Q1 = Value of 10+14th observation      = Value of 2.75th observation      = Value of 2nd observation + 0.75(Value of 3rd observa…

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