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Measures Based Upon Spread of Values

Range
Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus,
R = L - S

Higher value of Range implies higher dispersion and vice-versa.

Range: Comments
Range is affected by extreme values. As long as the minimum and maximum values remain unaltered, Range is not affected. Any change in other values does not affect range. It can not be calculated for open-ended frequency distribution. Range can be understood easily and used frequently because of its simplicity.

For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them.

Open-ended distributions are those in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified.

Quartile Deviation
The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, you need a measure which is not unduly affected by the outliers.

In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median. The upper and lower quartiles (Q3 and Q1, respectively) are used to calculate Inter Quartile Range which is Q3 - Q1. Inter-Quartile Range is based upon middle 50% of the values in a distribution and is, therefore, not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation. Thus:
Q .D. =
Q.D. is therefore also called Semi- Inter Quartile Range.
Calculation of Range and Q.D. for ungrouped data

Example 1
Calculate Range and Q.D. of the following observations:
20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70

Solution:
Range is clearly 70 - 20 = 50
For Q.D., we need to calculate values of Q3 and Q1.
Q1 is the size of value. Here, n being 11, Q1 is the size of 3rd value.
As the values are already arranged in ascending order, it can be seen that Q1, the 3rd value is 29. [What will you do if these values are not in an order?]
Similarly, Q3 is size of th value; i.e. 9th value which is 51. Hence Q3 = 51
Q.D = = = 11
Do you notice that Q.D. is the average difference of the Quartiles from the median.

Example 2
For the following distribution of marks scored by a class of 40 students, calculate the Range and Q.D.

Solution:
TABLE 6.1
 

Class intervals

No. of students

0-10

5

10-20

8

20-40

16

40-60

7

60-90

4

 

40


Range is just the difference between the upper limit of the highest class and the lower limit of the lowest class. So Range is 90 - 0 = 90. For Q.D., first calculate cumulative frequencies as follows:

C.I

Frequency

Cumulative Frequency

0-10

5

05

10-20

8

13

20-40

16

29

40-60

7

36

60-90

4

40

 

n

40

 

Q1 is the size of th value in a continuous series. Thus it is the size of the 10th value. The class containing the 10th value is 10-20. Hence Q1 lies in class 10-20. Now, to calculate the exact value of Q1, the following formula is used:


Where L = 10 (lower limit of the relevant Quartile class) c.f. = 5 (Value of c.f. for the class preceding the Quartile class) i = 10 (interval of the Quartile class), and f = 8 (frequency of the Quartile class) Thus,
= 16.25

Similarly,
Q3 is the size of th value; i.e., 30th value, which lies in class 40-60. Now using the formula for Q3, its value can be calculated as follows:



In individual and discrete series, Q1 is the size of th value but in a continuous distribution, it is the size of th value. Similarly, for Q3 and median also, n is used in place of n+1. If the entire group is divided into two equal halves and the median is calculated for each half, you will have the median of better students and the median of weak students. These medians differ from the median of the entire group by 13.31 on an average.

Similarly, suppose you have data about incomes of people of a town. Median income of all people can be calculated. Now if all people are divided into two equal groups of rich and poor, medians of both groups can be calculated. Quartile Deviation will tell you the average difference between medians of these two groups belonging to rich and poor, from the median of the entire group. Quartile Deviation can generally be calculated for open-ended distributions and is not unduly affected by extreme values.




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