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MEASURES OF DISPERSION

Made by :

BhanwarIshaan Sood

Jasmine Singh

Apandeep Singh


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Definition of Dispersion

Dispersion indicates the measure of the extent to

which individual items differ. It indicates lack of

uniformity in the size of items.

Dispersion or spread is the degree of the scatter or

variation of the variables about central value

OR

The degree to which numerical data tend to spreadabout an average value is called the Variation or

dispersion .


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Objectives of

Measuring Dispersion:


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It should be rigidly defined.

It should be simple to understand & easy

to calculate.

It should be based upon all values of given

data.It should be capable of further mathematical

treatment.

It should have sampling stability. It should be not be unduly affected by

extreme values.


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Absolute Measures of Dispersion:The measures of dispersion which are expressed in

terms of original units of a data are termed as

Absolute Measures.

Relative Measures of Dispersion:Relative measures of dispersion, are also known as

coefficients of dispersion, are obtained as ratios or

percentages.

These are pure numbers independent of the units

of measurement.


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Methods of Studying Dispersion:

1. Range

2. Quartile Deviation or Semi-inter quartile Range.

3. Mean Deviation.

4. Standard Deviation.


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RANGE

Definition:For Unclassified data: Range is defined as the

difference between the largest and the smallest values

of the data,

Symbolically,R = L S

Where L = Largest value, S = Smallest value, R = Range

The relative measure of range is defined as,


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RANGE

Definition:

For Classified data: Range is defined as the differencebetween the upper boundary of last class interval and

the lower boundary of first class boundary of the

distribution.

Symbolically,R = ULI LFI

Where ULI = upper boundary of last class interval,

LFI = lower boundary of first class interval, R = Range

The relative measure of range is defined as,


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MERITS OF RANGE :

1. Range is rigidly defined.2. Range is simple to understand and easy to calculate.

DEMERITS OF RANGE :

1. Range is not based upon all observation of given data.

2. Range is not capable for further mathematical treatment.

3. Range is much affected by extreme values.

4. Range is much affected by sampling variation.

5. Range can not be calculated for open end classes without

any assumptions.


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What is the range of the following data:

4 8 1 6 6 2 9 3 6 9

Soln: The largest score (L) is 9;

The smallest score (S) is 1;

Range= R =L - S = 9 - 1 = 8.

Coefficient of Range = R = = = = 0.8


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QUARTILE

DEVIATION

Definition:Quartile Deviation(Q.D.) is defined as

Q.D. =

Where Q3 = Upper (Third) quartile,

Q1 = Lower (First) quartile

The relative measure of quartile deviation is defined

as,

Coefficient of Q.D. =

13

13

QQ

QQ

2

13QQ


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MERITS OF QUARTILE DEVIATION :

1. Quartile deviation is rigidly defined.

2. Quartile deviation is simple to understand and easy tocalculate.

2. Quartile deviation is not affected by extreme values.

3. Quartile deviation can be calculated for open end classes

without assumptions

DEMERITS OF QUARTILE DEVIATION :

1. Quartile deviation is not based upon all observations of

data.

2. Quartile deviation is not capable of further mathematical

treatment.

3. Quartile deviation is much affected by sampling

fluctuations


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Range and Quartile deviation are not

based upon all observations. They are

positional measures of dispersion. They do not

show any scatter of the observations from an

average. The mean deviation based upon all

the observations.


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MEAN DEVIATION :

Definition:For Unclassified data: Let x1, x2,., xn are n observations of given

data. If n values x1, x2,., xn have an Arithmetic mean thenare the deviations of values from mean. Mean deviation about

mean is defined as follow,


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Similarly, If Me is median of given data, Then

Men deviation about median is given by,

If Mo is the mode of given data. Than Mean

Deviation about mode is,


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For Classified data: Let the variable X has values x1, x2,.,

xn with frequencies f1, f2,., fn

If n values x1, x2,., xn have an Arithmetic mean thenMean deviation about mean is defined as follow,


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Similarly, If Me is median of given data, Then Mean

deviation about median is given by,

If Mo is the mode of given data. Than Mean Deviation

about mode is,


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Merits of Mean Deviation

1. Mean deviation is rigidly defined.

2. Mean deviation is simple to understand and easy to calculate.3. Mean deviation is based upon all observations,

Demerits of Mean Deviation

1. The greatest drawback of Mean deviation is that algebraic signs

are ignored while taking deviations from items.

2. Mean deviation is not capable of further mathematical treatment.

3. Mean deviation is much affected by sampling variation.

4. Mean deviation is much affected by extreme values.5. There no hard & fast rule in the selection of particular average,

with respect to which the deviation are computed.

6. Mean deviation can not be calculated for open end classes

without any assumptions.


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Definition:

For Classified data: Let variables X has values x1,

x2,., xn with frequencies f1, f2,., fn . If n values x1,

x2,., xn have an Arithmetic mean Than Standarddeviation is given by

Where N = Total frequency


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STANDERD DEVIATION

Definition:

For Unclassified data: Let x1, x2,., xn are n

observations of given data. If n values x1, x2,., xn have

an Arithmetic mean Than

Standard deviation is given by


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Coefficient of variation (C.V.)

When this is expressed as percentage, that is multiplied by 100, it is

called Coefficient of variation. The coefficient of variation is the

ratio of standard deviation to the arithmetic mean expressed as

percentage.


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Merits of Standard deviation

1. Standard deviation is rigidly defined.

2. Standard deviation is based upon all observations.3. Standard deviation is capable of further mathematical

treatment.

4. Standard deviation is less affected by sampling

variations

Demerits of Standard deviation:

1. Standard deviation is not simple to understand

and not easy to calculate.

1. Standard deviation is much affected by extremevalues.

3. Standard deviation can not be calculated for

open end classes without any assumptions.

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