Dispersion Geeta Sukhija Associate Professor Department of Commerce Post Graduate Government College...
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Transcript of Dispersion Geeta Sukhija Associate Professor Department of Commerce Post Graduate Government College...
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.=