Dispersion Geeta Sukhija Associate Professor Department of Commerce Post 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 study To verify the reliability of an average Control on variability To know the variability Helpful in knowing the limit-range

Properties of a good measure of Dispersion Simple to understand Easy to compute Clear and stable in definition Rigidly defined Capable of further algebraic treatment Least affected by change in the sample Based on all the observations

Methods of measuring Dispersion Range Quartile deviation Mean Deviation Standard Deviation

Range It is the difference between the highest value and the lowest value in a series. R=H-L Range 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: Simple Used in Quality control Broad picture of data Demerits of Range Unstable Measure Not based on all items of series Range 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+Q1 2 = Q3-Q1 Q3+Q1

Mean Deviation It is based upon all items of the series It 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 – signs The 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 Merits Simple Less effect of extreme values It is useful in that series where we are interested in the study of mid part of series. Demerits Not based on all values Incomplete formation The calculation of QD is influenced by change in sample of population Limited 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 __ = ΣXΣX___ 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 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= Σx 2 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-X Each value of deviations is squared. The sum total of the square of deviations is obtained, Σx 2 Σx 2 is divided by the number of items in the series Square root of Σx 2 will be Standard Deviation. __ _ ___ N

Individual series Short cut Method Step Deviation Method ∑d 2 ∑d _ NN () 2 =S.D. ∑ ď 2 ∑ď NN _ () 2 X 100 S.D.=

Discrete Series Direct method Shortcut method ∑ f(X- X) 2 ‾ ______ N √ S.D.= ‾ ∑fd 2 N __ ( N ∑fd __ ) 2 _ √ _____________ S.D.=

Continuous Series Direct method Shortcut method ∑ f(X- X) 2 ‾ ______ N √ S.D.= ‾ ∑fd 2 N __ ( N ∑fd __ ) 2 _ √ _____________ S.D.=

Step deviation Method ∑fď2∑fď NN __ _ )( 2 ______________ √ S.D.=