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REU S HIGH SCHOOLStatistics project

Topic:- Measures of dispersionMade by:-Aanchal

lodha

Roll no :-21

Guided by:-Chacko

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INTRODUCTION

A modern student of statistics is mainly interested inthe study of variability and uncertainty. In this sectionwe shall discuss variability and its measures anduncertainty will be discussed in probability. We live in

a changing world. Changes are taking place in everysphere of life. A man of statistics does not show muchinterest in those things which are constant. The totalarea of the earth may not be very important to aresearch minded person but the area under different

crops, area covered by forests, area covered byresidential and commercial buildings are figures ofgreat importance because these figures keep onchanging form time to time and from place to place.

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CHARACTERISTICS OF MEASURES OF

DISPERSION

The defination should be clear.

It should be easy to compute.

It should be leat affected by samplingfluctuations.

All observations of the data should be used

in its calculation.

It should be amenable to further algebraic

manipulations.

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THE VARIOUS MEASURES OF DISPERSION ARE

AS FOLLOW:

1. Range

2. Quartile deviation

3. Mean deviation4. Standard deviation

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RANGE

In any statistical series, the difference

between the largest and the smallest values

is called as the range.

Thus Range (R) = LS

where,

L=largest value of the series

S=Smallest value of the series

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COEFFICIENT OF RANGE

The relative measure of the range. It is used in thecomparative study of the dispersion co-efficient of Range= L-S/L+S

110, 117, 129, 197, 190, 100, 100, 178, 255, 790. Solution: R = L - S = 790 - 100 = 690

Co-efficient of Range = L-S/L+S= 790-100/790+100=0.78

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MEAN DEVIATION

In the defination of range and Q.D. as

meaures of dispersion the use of diffrences

of observations of the data from the mean is

not considered. These diffrences are calledmean deviation

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STANDARD DEVIATION AND VARIANCE

By far the most commonly used measures of dispersion inthe social sciences are varianceand standarddeviation. Varianceis the average squared difference ofscores from the mean score of a distribution. Standarddeviationis the square root of the variance.

The standard deviationis simply the square root of thevariance. In some sense, taking the square root of thevariance "undoes" the squaring of the differences that wedid when we calculated the variance.

Variance and standard deviation of a populationaredesignated by and , respectively. Variance and standarddeviation of a sampleare designated by s2and s,respectively.

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VARIANCE S D

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MEAN DEVIATION

It is the average of the modulus of the deviations of the observations in a series

taken form mean or median.

Methods for Calculation of Mean Deviation:

Case I:For Ungrouped Data.

In this case the mean deviation is given by the formula

Mean Deviation = M.D. = |x- A|/n = |d|/n,

Where d stands for the deviation from the mean or median and |d| is always

positive whether d itself is positive or negative and n is the total number of items.

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CONTINUE..

Case II: For Grouped data.

Let x1, x2, x3, , xn occur with frequencies f1, f2, f3, ,fnrespectively andlet f= n and M can be either Mean or Median, then the mean deviationis given by the formula.

Mean Deviation = f|x-M|/f= f|d|/n

Where d = |xM| and f= n.

Coefficient of Mean Deviation = Mean Deviation / Median

or = Mean Deviation / Mean (In case the deviations are taken from mean)

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QUARTILE DEVIATION:

The quartile deviation is a slightly better measure

of absolute dispersion than the range. But it

ignores the observation on the tails.

Quartile deviation = Q3- Q1/ 2

Coefficient of Quartile Deviation = Q3- Q1/ Q3-Q1

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