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Lecture.5

Measures of dispersion - Range, Variance -Standard deviation - co-efficient of

variation - computation of the above statistics for raw and grouped data

Measures of Dispersion

The averages are representatives of a frequency distribution. But they fail to give

a complete picture of the distribution. They do not tell anything about the scatterness of

observations within the distribution.

Suppose that we have the distribution of the yields (kg per plot) of two paddy

varieties from 5 plots each. The distribution may be as follows

Variety I 45 42 42 41 40

Variety II 54 48 42 33 30

It can be seen that the mean yield for both varieties is 42 kg but cannot say that

the performances of the two varieties are same. There is greater uniformity of yields in

the first variety whereas there is more variability in the yields of the second variety. The

first variety may be preferred since it is more consistent in yield performance.

Form the above example it is obvious that a measure of central tendency alone is

not sufficient to describe a frequency distribution. In addition to it we should have a

measure of scatterness of observations. The scatterness or variation of observations from

their average are called the dispersion. There are different measures of dispersion like the

range, the quartile deviation, the mean deviation and the standard deviation.

Characteristics of a good measure of dispersion

An ideal measure of dispersion is expected to possess the following properties

1. It should be rigidly defined

2. It should be based on all the items.

3. It should not be unduly affected by extreme items.

4. It should lend itself for algebraic manipulation.

5. It should be simple to understand and easy to calculate

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Range

This is the simplest possible measure of dispersion and is defined as the difference

between the largest and smallest values of the variable.

? In symbols, Range = L - S.

? Where L = Largest value.

? S = Smallest value.

In individual observations and discrete series, L and S are easily identified.

In continuous series, the following two methods are followed.

Method 1

L = Upper boundary of the highest class

S = Lower boundary of the lowest class.

Method 2

L = Mid value of the highest class.

S = Mid value of the lowest class.

Example1

The yields (kg per plot) of a cotton variety from five plots are 8, 9, 8, 10 and 11. Find the

range

Solution

L=11, S = 8.

Range = L - S = 11- 8 = 3

Example 2

Calculate range from the following distribution.

Size: 60-63 63-66 66-69 69-72 72-75

Number: 5 18 42 27 8

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Solution

L = Upper boundary of the highest class = 75

S = Lower boundary of the lowest class = 60

Range = L - S = 75 - 60 = 15

Merits and Demerits of Range

Merits

1. It is simple to understand.

2. It is easy to calculate.

3. In certain types of problems like quality control, weather forecasts, share price

analysis, etc.,

range is most widely used.

Demerits

1. It is very much affected by the extreme items.

2. It is based on only two extreme observations.

3. It cannot be calculated from open-end class intervals.

4. It is not suitable for mathematical treatment.

5. It is a very rarely used measure.

Standard Deviation

It is defined as the positive square-root of the arithmetic mean of the Square of the

deviations of the given observation from their arithmetic mean.

The standard deviation is denoted by s in case of sample and Greek letter

(sigma) in case of population.

The formula for calculating standard deviation is as follows

for raw data

And for grouped data the formulas are

3

What is the difference between standard deviation and variation?Variance: Whatâ€™s the Difference?Standard Deviation. Standard deviation is a statistic that looks at how far from the mean a group of numbers is, by using the square root of the variance.Variance. The variance is the average of the squared differences from the mean. ...Standard Deviation and Variance in Investing. ...The Bottom Line. ...

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