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Measures of

Dispersion


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Defination

While measures of central tendencyindicate what value of avariable is (in one sense or other) average or central or

typical in a set of data, measures of dispersion(or variabilityorspread) indicate (in one sense or other) the extent to which theobserved values are spread out around that center how far

apart observed values typically are from each other or from someaverage value (in particular, the mean). Thus: if all cases have identical observed values (and thereby are

also identical to [any] average value), dispersion is zero; if most cases have observed values that are quite close

together (and thereby are also quite close to the averagevalue), dispersion is low (but greater than zero); and

if many cases have observed values that are quite far awayfrom many others (or from the average value), dispersion ishigh.


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Measures of Dispersion

Synonym for variability

Often called spread or scatter

Indicator of consistency among adata set

Indicates how close data areclustered about a measure ofcentral tendency


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Example

Consider the following data related to agedistribution of two groups A and B:

avg

Grp A 22 24 25 26 28 25

Grp B 8 15 20 28 54 25


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Above mentioned two groups have thesame average i.e. 25 years, so we arelikely to conclude that the two groups are

similar. Wrong conclusion as the obs. in group A

are close to one another indicating thatpeople in this group are more or less of

the age 22 to 28 years.


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While those in group B are widely dissimilar andhave greater variability of ages as it includes aperson who is 8 years old on one hand and aperson of age 54 on the other hand.

This means that central value does not give theclear indication of the pattern of distribution.

Measure of dispersion or variability gives us theinformation about the spread of the obs. In one

distribution

Here, dispersion of group B is more than that ofgroup A


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Purpose of Measuring Variation

To test the reliability of an average

To serve as a basis for control ofvariability

To compare two or more series withregard to variability

To facilitate as a basis for furtherstatistical analysis.


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Properties of a good measure of

variation

It should be simple to understand and easy to calculate.

It should be based on all observations.

It should be amenable to further algebraic treatment.

It should not be affected by extreme observations.


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Measures of variation

Absolute measures

Range

Quartile deviation

Mean deviation

Standard deviation / variance

Lorenz curve

Relative measures

Coefficient of range

Coefficient of variation

Coefficient of quartile deviation

Coefficient of mean deviation


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Absolute measures of variation

They are expressed in the same statistical unit inwhich the original data are given such as rupees,kg etc.

These values are used to compare the variationin two or more than two distributions providedthe variables are expressed in the same units andhave almost the same average value.


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Relative measures of variation

Absolute measure of dispersion expressesvariation in the same units as the originaldata

To compare the variations of two differentseries, relative measure of standarddeviation is calculated.


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Range

Range is the preliminary indicator of dispersion.

The (total or simple) range is the maximum(highest) value observed in the data [the value ofthe case at the 100th percentile] minus the

minimum(lowest) value observed in the data[the value of the case at the 0th percentile] That is, it is the distance or interval between the

values of these two most extreme cases.

Indicates how spread out the data are.

Open-ended distributions have no range bz nohighest or lowest values exist in an open-endedclass.


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The Range

The rangeis defined as the differencebetween the largest score in the set ofdata and the smallest score in the set of

data, XL- XS What is the range of the following data:

4 8 1 6 6 2 9 3 6 9

The largest score (XL) is 9; the smallest

score (XS) is 1; the range is XL- XS= 9 -1 = 8


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Coefficient of scatter

Ratio of range

Coefficient of range =(Max- Min )/ (Max +Min) = Absolute range / Sum of the

extreme values


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Dispersion Example

Number of minutes 20clients waited to see aconsultant

ConsultantX Y

05 15 11 12

12 03 10 13

04 19 11 1037 11 09 13

06 34 09 11

Consultant X:

Sees some clientsalmost immediately

Others wait over 1/2hour

Highly inconsistent

Consultant Y:

Clients wait about 10

minutes 9 minutes least wait and

13 minutes most

Highly consistent


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Solution

1.Coefficient of range

=(Max- Min )/ (Max + Min)

= (37- 03 )/ (37 + 03) = 34/40 = 0.85

2. Coefficient of range

=(Max- Min )/ (Max + Min)= (13- 09 )/ (13 + 09) = 4/22 = 0.18

Consultant X is inconsistent and Consultant Y is consistent intheir job.


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Uses

QUALITY CONTROL: The objective of quality control is to keep a check on the

quality of the product without 100% inspection

When statistical methods of quality control are used,control charts are prepared in which range plays animportant role.

The basic idea is that as long as manufactured productsconform to set standards (range), the productionprocess is assumed to be in control.

WHEATHER FORECASTS: This helps the general public to know as to what limits

the temperature is likely to vary on a particular day.


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Quartile deviation

It measures the distance between thelowest and highest of the middle 50percent of the scores of distribution.

Q.D. is superior to range, as it is notbased on two extreme values but ratheron middle 50% observation.

It can be calculated from open-ended

classes. It is often used with skewed data as it is

insensitive to the extreme scores


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Interquartile Range

Interquartile range = Q3Q1

Semi-interquartile range or quartile

deviationis defined as

= (Q3Q1)/2

Coefficient of quartile deviationis

= = (Q3Q1)/(Q3+ Q1)


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When Q.D. is small then it describes highuniformity of central 50% observations.

High Q.D. means high variation among the

central observations.


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Interquartile Range Example

The number of complaints received by themanager of a supermarket was recorded foreach of the last 10 working days.

21, 15, 18, 5, 10, 17, 21, 19, 25 & 28

Sorted data

5, 10, 15, 17, 18, 19, 21, 21, 25 & 28

nObservatioorQ

Q

nQ

rd375.2

4

11

4

1

1

1

1

nObservatioorQ

Q

nQ

th825.8

4

33

4

13

3

3

3

Interquartile range = 21

15 = 6 days


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Calculating exactly:Q1

Using the formula:

16

X f CF

0 < 20 15 15

20 < 40 60 75

40


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Q3

17

Third QuartileThis is in the group 20 < 40

Lower limit (l) is 20

Width of group (i) is 20

Frequency of group (f) is 60CF of previous group (F) is 15

X f CF

0 < 20 15 15

20 < 40 60 75

40


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Interquart e Range an Coe c ent o Q.D.

Interquartile range = 40-23.333= 16.671

Semi-interquartile range or quartile

deviationis defined as= (Q3Q1)/2 = 16.67/2 =8.335

Coefficient of quartile deviationis= = (Q3Q1)/(Q3+ Q1) = 16.67/ 63.33

= 0.26


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ExampleWeekly income (Rs.) no. of workers

below 1350 8

1350-1370 16

1370-1390 39

1390-1410 58

1410-1430 60

1430-1450 40

1450-1470 22

1470-1490 15

1490-1510 15

1510-1530 9

1530 and above 10

Use an appropriatemeasure toevaluate thevariation in thefollowing data:


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Problems with quartile Deviation

It is not based on all the observations

Affected by sampling fluctuations

Not suitable for further algebraic treatment


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Deviation Measures of Dispersion (cont.) The deviation from the mean for a representative case iis

xi- mean ofx.

If almost all of these deviations are small, dispersion is small. If many of these deviations are large, dispersion is large.

This suggests we could construct a measure Dof dispersionthat would simply be the average (mean) of all thedeviations.

But this will not work because, as we saw earlier, it is a

property of the mean that all deviation from it add up to

zero.


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DeviationMeasuresof Dispersion: Example(cont.)


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The Mean Deviation A practical way around this problem is simply to ignore the

fact that some deviations are negative while others are

positive by averaging the absolute valuesof the deviations. This measure (called the mean deviation) tells us the

average(mean) amount that the values for all casesdeviate(regardless of whether they are higher or lower)from the average(mean) value.

Indeed, the Mean Deviation is an intuitive, understand-able, and perfectly reasonable measure of dispersion, and itis occasionally used in research.

The mean deviation takes into consideration all of thevalues.


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The Mean Deviation (cont.)


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If the data are in the form of a frequencydistribution, the mean deviation can be calculatedusing the following formula:

Where: f= the frequency of an observation x

n = f= the sum of the frequencies

This measure is an improvement over theprevious two measures in the sense that itconsiders all observations of a data set.

Frequency Distribution Mean Deviation

f

xxfMD

_

||


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Coefficient of mean deviation

Coefficient of mean deviation =

= Mean deviation

Mean

E l


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Example

Find out the mean deviation for the following distribution of

demand for a bookQuantity

demanded(in unit)

Frequency fx |x-x| f|x-x|

6 4 24 17.6 70.4

12 7 84 11.6 81.2

18 10 180 5.6 56

24 18 432 0.4 7.2

30 12 360 6.4 76.8

36 7 254 12.4 86.8

42 2 84 18.4 36.8

total 60 fx = 1416 f|x-x| =415.2

mean =

1416/60=23.6

MD=

415.2/60=6.92

f

xxfMD

_

||

f

fxx_

x


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Problems with Mean Deviation

Algebraic signs are ignored while takingthe deviations of the items.

Cannot be computed for distribution

with open end classes.

Not suitable for further mathematicaltreatment.


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Standard Deviation

Standard deviation is the most commonlyused measure of dispersion

Similar to the mean deviation, the

standard deviation takes into account thevalue of every observation

It is the measure of the degree ofdispersion of the data from the meanvalue.


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First, it says to subtract the mean fromeach of the scores

This difference is called a deviateor a

deviation score The deviate tells us how far a given score is

from the typical, or average, score

Thus, the deviate is a measure of dispersion

for a given score


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It is a static that tells us how tightly allthe various values are clustered aroundthe mean in set of data.

Large S.D. indicates that data points arefar from the mean

Small S.D. indicates that all the datapoints cluster closely around the mean.


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Standard Deviation It is the positive square root of thearithmetic mean of the squares of the

deviations of the observations from theirarithmetic mean.

Calculation:

Calculate the arithmetic mean (AM) Subtract each individual value from the AM Square each value -- multiply it times itself Sum (total) the squared values Divide the total by the number of values (N)

Calculate the square root of the value

Formula:

n

xx

2_


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The Mean, Deviations, Variance, and SD

What is the effect of adding a constant amount to (orsubtracting from) each observed value?

What is the effect of multiplying each observed value (ordividing it by) a constant amount?


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) Adding (subtracting) the same amount to(from) every observed value changes themean by the same amount but does not

change the dispersion (for either range ordeviation measures

Multiplying every observed value by thesame factor changes the mean and the SD

[or MD] by that same factor and changesthe variance by that factor squared.


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usefulness

Manufacturers interested in producing items ofconsistent quality are very much concerned withS.D.

If the mean life of the component is 4 years and

the S.D. is very large, it would correspond tomany failures large before 4 years.

Quality control requires consistency andconsistency requires a relatively small S.D.

V i


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The square of the standard deviation.More useful when we begin analysis ratherthan description:

1

)( 22

n

xxs

Variance

What Does the Variance Formula


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What Does the Variance Formula

Mean?

Variance is the mean of the squareddeviation scores

The larger the variance is, the more the

scores deviate, on average, away from themean

The smaller the variance is, the less thescores deviate, on average, from the

mean


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Combined Variance (For different means)

21

2

2

2

22

2

1

2

11 )()(

nn

dndn


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Exercise 3

The mean and s.d of the lives of tyres ofmanufactured by two factories of Durable tyrecompany, making 50,000 tyres annually , at eachof the two factories , are given below. Calculate

combined mean and standard deviation of thelife of all the 100000 tyres produced in a year.

Factory Sample Size Mean (000 Kms) SD(000 Kms)

1 50 60 82 50 50 7


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Combined Variance (For same means)

21

2

22

2

11nnnn


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Example

The following data isrelated to clientsobtained by insuranceagents during a given

period for two types ofinsurance policies, achild policy and aretirement policy.

Calculate thecombined S.D.

Child

policy

Retirem

entpolicy

No. ofagents

25 18

Averageno. ofclientsbooked

72 64

Variance

of thedistribution

8 6


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The Coefficient of Variation

It is the most important relative measures ofdispersion

One ratio measure of dispersion/inequality is the coefficientof variation, which is simply the standard deviation divided

by the mean. It answers the question: how big is the SD relative to

the mean?

100variationoftcoefficien

x

s


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It is therefore a useful statistic to compare thedegree of variation from one data series toanother.

It helps us to determine how much volatility

(risk) we are assuming in comparison to theamount of return one can expect from aninvestment

Lower the coefficient of variation, better the risk-return tradeoff.

The distribution for which C.V. is more is said tobe less stable, less uniform, less consistent, lesshomogeneous.


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Measure of Skew

Skewis a measure of symmetry in thedistribution of scores

Positive

Skew

Negative Skew

Normal(skew = 0)


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Measure of Skewness

Measure of skewness of a distribution isgiven by

=3(mean median)

S.D.This measure is known as Karl Pearsons

coefficient of skewness and lies b/w -3and +3.


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A distribution is said to be symmetric if mean =median = mode

A distribution is said to be positively skewed if

mean > median > mode

A distribution is said to be negatively skewed ifmean < median < mode

The smaller the number- the less the skewness.If co.skew=0 then the data is exactly balanced.

Bell -Shaped Curve showing the relationship between and . m


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m m2 m1 m m 1 m 2 m 3

p g p m

68%

95%

99.7%

0actual dispersion

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