Dispersion

Geeta SukhijaAssociate Professor

Department of CommercePost Graduate Government College for Girls

Sector 11, Chandigarh

Meaning of Dispersion

Dictionary Meaning: Spread, scatteredness or variation

Dispersion refers to the variation of the around an average or among themselves.

The word Dispersion is therefore used in two different ways:

The extent of variability in a given data is measured by taking the difference of highest value and the lowest value. It is called averages of first order. Here dispersion is the variation of the values of items of a series among themselves.

Meaning and definition of Dispersion

There is second meaning of dispersion also. It also refers too the variation of the items around an average. If values of a series are widely different from their average, it would mean a higher degree of dispersion. Such measure is called average of second order. It is this meaning of dispersion that is widely used by statisticians.

Dr Bowley: Dispersion is the measure of the variation of the items.

Cannor: Dispersion is a measure of the extent to which the individual items vary.

Importance of Dispersion

Comparative studyTo verify the reliability of an averageControl on variabilityTo know the variabilityHelpful in knowing the limit-range

Properties of a good measure of Dispersion

Simple to understandEasy to computeClear and stable in definitionRigidly definedCapable of further algebraic treatmentLeast affected by change in the sampleBased on all the observations

Methods of measuring Dispersion

RangeQuartile deviation

Mean DeviationStandard Deviation

Range

It is the difference between the highest value and the lowest value in a series.

R=H-LRange is the absolute measure of dispersion,

it can not be used for comparison. To make it comparable, we find its coefficient.

Coefficient of range:C R=

H-L

H+L

Merits and demerits of Range

Merits:SimpleUsed in Quality controlBroad picture of data

Demerits of RangeUnstable MeasureNot based on all items of seriesRange gives no knowledge about the formation of the

series.Range depends on extreme values of the series. So it

is affected when the sample changes.It can not be calculated in case of open ended series.

Inter-quartile range and Quartile Deviation

The range is based on extreme values. It ignores the deviation in between values. In order to study variation among values, we study Inter-quartile range

Inter-quartile range is the difference between the third quartile and the first quartile.

Inter-quartile range=Q3-Q1

Quartile Deviation=Q3-Q1

2

.

Coefficient of Q.D. = Q3-Q1

2÷

Q3+Q12

=Q3-Q1

Q3+Q1

Mean Deviation

It is based upon all items of the seriesIt is the arithmetic average of the deviation of all the

values taken from some average value( mean, mode, median) of the series, ignoring signs ( + or - ) of the deviations.

The deviations are generally taken from median.These deviations are summed up ignoring their + or

– signsThe deviations are noted as |d| and sum of deviations

as Σ |d| is taken. The sum of deviations is divided by the number of items to find the mean deviation.

MD= Σ |d|N

Merits and Demerits of Q.D

MeritsSimpleLess effect of extreme valuesIt is useful in that series where we are

interested in the study of mid part of series.Demerits

Not based on all valuesIncomplete formationThe calculation of QD is influenced by change

in sample of populationLimited use

Coefficient of Mean Deviation

Coefficient of MD from Mean=

Coefficient of MD from Median=

Coefficient of MD from Mode=

MDx

X

MD m

M

MDz

Z

Calculation of Mean Deviation

Mean Deviation

Individual Series

Discrete Series

Continuous Series

Calculation of Mean Deviation from Median

Calculation of Mean Deviation from Mean

M= Size of N+1 th item

MDm=Σ|dm|

Coefficient of MDm= MDm

X MD x= Σ|dm|

Coefficient of MD x =

MD x

Mean deviation: Individual Series

____2

____N

______M

__ = ΣX___

N

_____N

_______

X __

__

__

Mean deviation: Discrete Series

Steps:Find out Mean or median from which

deviations are to be taken out.Deviation of different items in the series are

taken from mean/median, and signs of + or – of the deviations are ignored.

Each deviation value is multiplied by the frequency facing it and sum of the multiples is obtained.

Thus MDm= Σf|dm|____

N

Mean deviation: Continuous Series

Steps:Continuous series are first converted into

discrete series by finding mid values of the class interval.

The same procedure is used for calculation of mean deviation and its coefficient as in case of discrete series.

Standard Deviation

SD is a precise measure of dispersion.This concept was introduced by Karl Pearson

in 1893.It is also called Mean Error or Mean Square

Error or Root-Mean Square Deviation.SD is the positive square root of the

arithmetic mean of the square of deviations of the items from their mean value.

SD= Σx2 or Σ(X-X)___

N

_ 2

_____N

Calculation of Standard Deviation

Coefficient of SD=S.D.___

X_

Individual Series

Direct Method

Short cut method

Step Deviation Method

Individual series: Direct Method

First find out Mean value of series (X)Deviation of each item from mean is

determined ie We find out x=X-XEach value of deviations is squared.The sum total of the square of deviations is

obtained, Σx2 Σx2 is divided by the number of items in the

seriesSquare root of Σx2 will be Standard Deviation.

__

_

___N

Individual series

Short cut Method

Step Deviation Method

∑d2 ∑d_

N N( )2

=S.D.

∑ ď 2 ∑ď

N N

_( )2

X 100S.D.=

Discrete Series

Direct method

Shortcut method

∑ f(X- X)2‾______

N√S.D.=

‾∑fd2

N__ (

N

∑fd__ )

2_√

_____________

S.D.=

S.D.=

Continuous Series

Direct method

Shortcut method

∑ f(X- X)2‾______

N√S.D.=

‾∑fd2

N__ (

N

∑fd__ )

2_√

_____________

S.D.=

Step deviation Method

∑fď2 ∑fď

N N

__ ___ )(2

______________

√S.D.=