Measure of Dispersion in statistics
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Transcript of Measure of Dispersion in statistics
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Measure of Dispersion
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We are Group 4
Tasnim Ansari Hridi (ID-09)
Md. Mehedi Hassan Bappy (ID-21)
Debanik Chakraborty (ID-25)
Syed Ishtiak Uddin Ahmed (ID-31)
Devasish Kaiser (ID-49)
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Definition of Measure of Dispersion
In statistics, dispersion (also called
variability, scatter, or spread) is the extent
to which a distribution is stretched or
squeezed. Common examples
of measures of statistical dispersion are the
variance, standard deviation, and
interquartile range.
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Example
Centre: Same
Variation: Different
Year 2000: Close Dispersion
Year 2015: Wide Dispersion
Better Quality Data: Data of Year 2000
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Why Measure of Dispersion
Serve as a basis for the
control of the variabilityTo compare the variability
of two or more series
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Facilitate the use of other
statistical measures.
Reliable
Determine the reliability of
an average
Why Measure of Dispersion
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Characteristics of an Ideal
Measure of Dispersion
Must be based on all observations of the data.
It should be rigidly defined
It should be easy to understand and calculate.
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Must be least affected by the sampling
fluctuation.
Must be easily subjected to further mathematical
operations
Characteristics of an Ideal
Measure of Dispersion
It should not be unduly affected by the extreme
values.
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Types of Measures of
Dispersion
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Classification of
Measures of dispersion
in Statistics
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Measures of
Dispersion
Algebraic
Absolute Relative
Graphical
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Algebraic Measure of Dispersion
ร Mathematical way to calculate the
measure of dispersion.
Example: Calculation of Standard Deviation
or Co-efficient of Variance by using numbers
and formulas.
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Characteristics of Algebraic
Measure of Dispersion
โข Mathematical Way
โข Algebraic Variables are used
โข Numerical Figures are used here
โข Formulas & Equations are used
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Graphical Measure of Dispersion
ร The way to calculate the measure of
dispersion by figures and graphs.
Example: Calculation of Dispersion among
the heights of the students of a class from
the average height using a graph.
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Characteristics of Graphical
Measure of Dispersion
โข It is a visual way of measuring dispersion
โข Graphs, figures are used
โข Sometimes, it cannot give the actual result
โข It helps the reader to have an idea about the
dispersion practically at a glance
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Absolute Measure of Dispersion
Absolute Measure of Dispersion gives an idea about the
amount of dispersion/ spread in a set of observations. These
quantities measures the dispersion in the same units as the
units of original data. Absolute measures cannot be used to
compare the variation of two or more series/ data set.
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Classification of
Algebraic Measure of
Dispersion
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Absolute Measure of
Dispersion
Absolute Measure of Dispersion gives an idea about the
amount of dispersion/ spread in a set of observations. These
quantities measures the dispersion in the same units as the
units of original data. Absolute measures cannot be used to
compare the variation of two or more series/ data set.
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Relative Measure of Dispersion
These measures are a sort of ratio and are called coefficients.
Each absolute measure of dispersion can be converted into
its relative measure.
It can be used to compare two or more data sets
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Difference Between Absolute and Relative Measure of
Dispersion
3
This is calculated from original dataThese measure are calculated absolute
measures
2
It is not expressed in terms of percentage It is expressed in terms of percentage
1
It has the variable unit It has no unit
Absolute Measure Relative Measure
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6
There is no change in variables and with the absolute measures.
There is changes in variables with relative measures.
5
These measure cannot be used to compare the variation of two or more series
These measure can be used to compare the variation of two or more series.
4
No use of ratio Use of ratio
Absolute Measure Relative Measures
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Absolute measures of Dispersion
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Classification of Absolute measure
Mean Deviation
Quartile Deviation Standard Deviation
Range
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โRange
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Range
The difference between the maximum and
minimum observations in the data set.
R= H-L
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5, 10 , 15 , 20, 7, 9, 12 , 17 , 13 , 6 , 10 , 11
, 17 , 16
Range = H- L
= 20- 5 = 15
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Merits and Demerits of Range
Gives a quick answer
Cannot be calculated in open ended
distributions
Affected by sampling fluctuations
Changes from one sample to the
next in population
Gives a rough answer and is not
based on all observationSimple and easy to
understand
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โMean deviation
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Mean deviation
The average of the absolute values of
deviation from the mean(median or mode) is
called mean deviation.
=๐ | ๐โ๐ |
๐ต
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Merits of Mean deviation
Simplifies calculations
Can be calculated by mean, median
and mode
Is not affected by extreme measures
Used to make healthy
comparisons
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Demerits of Mean Deviation
Not reliable
Mathematically illogical to assume all
negatives as positives
Not suitable for comparing
series
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โQuartile Deviation
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Quartile Deviation
The half distance
between 75th
percentile i.e. 3rd
quartile (Q1) and 25th
percentile i.e. 1st
quartile (Q3) is
Quartile deviation or
Interquartile range.
Q.D = Q3 โ Q1
๐
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Has better result than
range mode.
Is not affected by
extreme items
Merits of Quartile Deviation
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Demerits of Quartile Deviation
It is completely dependent on the central items.
All the items of the frequency distribution are not given equal importance in finding the values of Q1 and Q3
Because it does not take into account all the items of the series, considered to be inaccurate.
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โStandard Deviation
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Standard Deviation
Standard deviation is calculated as the
square root of average of squared
deviations taken from actual mean.
It is also called root mean square
deviation.
= โ ๐โ ๐
๐
๐
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68.2%
95.4%
99.7%
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Merits of standard deviation
It takes into account all the items and is capable of future algebraic treatment and statistical analysis.
It is possible to calculate standard deviation for two or more series
This measure is most suitable for making comparisons among two or more series about variability.
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Demerits of Standard Deviation
It is difficult to compute. It assigns more
weights to extreme items and less
weights to items that are nearer to
mean.
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Classifications ofRelative Measures of
Dispersion
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Chart of classification
Relative Measure
Coefficient of Range
Coefficient of Quartile
Deviation
Coefficient of Mean
Deviation
Coefficient of Variation
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Coefficient of
Range
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Coefficient of Range
The measure of the distribution based on range
is the coefficient of range also known as range
coefficient of dispersion.
Formula:
Coefficient of Range= ๐ ๐๐๐๐
๐ป๐๐โ๐๐ ๐ก ๐๐๐๐ข๐+๐ฟ๐๐ค๐๐ ๐ก ๐ฃ๐๐๐ข๐ร 100
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โCoefficient of
Quartile Deviation
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Coefficient of Quartile Deviation
A relative measure of dispersion based on the
quartile deviation is called the coefficient of
quartile deviation.
Formula:
Coefficient of Quartile Deviation = ๐๐ข๐๐๐ก๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐๐๐๐๐๐ร 100
= Q3 โ Q1
Q3 + Q1
ร 100 [By Simplification]
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Merits & Demerits of Coefficient of Quartile
Deviation
Merits
1. Easily understood
2. Not much Mathematical
Difficulties
3. Better Result than
Coefficient of Range
Sampling fluctuation
Ignorance of last 25% of data sets.
Values being irregular
Demerits
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Coefficient of
Mean Deviation
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Coefficient of Mean Deviation
The relative measure of dispersion we get by dividing
Mean Deviation by Mean or Median, is called Coefficient
of Mean Deviation.
Formula:
Coefficient of MD= ๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐๐๐๐๐๐ ๐๐ ๐๐๐๐ร 100
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Merits & Demerits of Coefficient of Mean
Deviation
Merits
1. Better Result than Range
& Quartile Coefficient.
2. Least sampling fluctuation.
3. Rigidly defined.
Fractional Average.
Cannot be used for
sociological studies
Less reliable than
Coefficient of Variation
Demerits
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Coefficient of
Variation
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Coefficient of Variation
Coefficient of Variation is a measure of spread
that describes the amount of variability relative to the mean.
Formula:
Coefficient of Variation= ๐๐ก๐๐๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐๐๐๐ร 100
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Merits & Demerits of Coefficient of Variation
Merits
1. Best one
2. Most appropriate one
3. Based on Mean and
Standard Deviation
4. COV is dimensionless or non-
unitized
It is impossible to calculate if
Mean is 0
It is difficult to calculate if
the values are both positive
and negative numbers & if
the mean is close to 0.
Demerits
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Practical Uses of Coefficient of Variance
INVESTMENT ANALYSIS
STOCK MARKET
RISK EVALUATION
COMBINED STANDARD DEVIATION OF SEVERAL GROUPS
PERFORMANCES OF TWO PLAYERS
INDUSTRIES & FACTORIES
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MathematicalApplication
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Coefficient of range
Let 1,2,4,6,7 is a set of values of a distribution. Here, Highest Value, XH=7 & Lowest Value, XL=1 So, Range, R= 7-1 = 6Now, Coefficient of Range = ๐
XH + XL ร 100
= ๐
๐+๐ร 100 =75%
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Coefficient of Quartile
deviation Let the number of students in 5 classes are 110, 150, 180, 190, 240 is a set of values. Here, Q1= size of ๐+๐
๐th item = 130
And, Q3 = size of ๐(๐+๐)๐
th item = 215
So, Coefficient of Quartile Deviation =Q3 โ Q1Q3 + Q1
ร 100
= 215โ130215+130 ร 100= 24.64 %
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Coefficient of Mean Deviation
Let the ages of 5 boys in a class is 12, 14, 14, 15, 18.So their Mean, ๐ฑ = ๐๐+๐๐+๐๐+๐๐+๐๐
๐= 14.6
Mean Deviation, MD = | ๐ โ ๐ |
๐ต
=|12โ14.6| + |14โ14.6| + |14โ 14.6|+ |15โ14.6| + |18โ14.6|๐
= 1.52
Now, the Coefficient of MD= ๐๐
๐ฑร ๐๐๐ = ๐.๐๐
๐๐.๐ร ๐๐๐ = 10.41%
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Coefficient of
Variation Suppose the returns on an investment for 4 years is Tk.1000, Tk.3000, Tk.4500 & Tk.5000.
So, Mean, ๐ฑ = 3375 Standard Deviation, SD = 1796.99
So,Coefficient of Variation, CV= ๐๐
๐ฑร 100
= ๐๐๐๐.๐๐๐๐๐๐
ร 100 = 53.24%
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The daily sale of sugar in a certain grocery shop is given below : Monday Tuesday Wednesday Thursday Friday Saturday 75 kg 120 kg 12 kg 50 kg 70.5 kg 140.5 kg respectively.
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โ
No of Days sale of sugarMonday 60Tuesday 120Wednesday 10Thursday 50Friday 70Saturday 140
๐ฎ ๐๐ ๐ซ๐๐๐ = ๐ ๐ฎ๐ = ๐๐๐
Mean, ๐ฅ = ๐ฅ
๐=
4๐๐
6= 7๐
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โ
x ๐๐
60 3600120 1440010 10050 250070 4900
140 19600๐ฎ๐ = ๐๐๐ ๐ฎ๐๐= 45100
Standard deviation: ๐ =๐ฎ๐๐
๐โ
๐ฎ๐
๐
๐=
๐๐๐๐๐
๐โ
๐๐๐
๐
๐=
๐๐๐๐. ๐๐ โ ๐๐๐๐ = ๐๐. ๐๐
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Quartile Deviation
The marks of 7 students in Mathematics result are given below :
70, 85, 92,68, 75, 96, 84Find out-
โข First Quartile Deviationโข Third Quartile Deviation
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Quartile deviation
ร First quartile
๐๐ = ๐ฌ๐ข๐ณ๐ ๐จ๐๐ง + ๐
๐
๐ญ๐ก
๐ข๐ญ๐๐ฆ
= size of ๐+๐
๐
๐ญ๐ก๐ข๐ญ๐๐ฆ
= size of 2nd item.= 70
รThird Quartile
๐ธ๐ = ๐๐๐๐ ๐๐๐ ๐ + ๐ ๐๐
๐๐๐๐๐
= size of ๐ ๐+๐ ๐๐
๐๐๐๐๐
= size of 6th item=92
Arranging the data in ascending order we get,68,70,75,84,85,92,96
![Page 65: Measure of Dispersion in statistics](https://reader034.fdocuments.in/reader034/viewer/2022050614/5a6d2c5a7f8b9aff418b4ebb/html5/thumbnails/65.jpg)
โThank you