Download - Measure of Dispersion in statistics
Measure of Dispersion
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)
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.
Example
Centre: Same
Variation: Different
Year 2000: Close Dispersion
Year 2015: Wide Dispersion
Better Quality Data: Data of Year 2000
Why Measure of Dispersion
Serve as a basis for the
control of the variabilityTo compare the variability
of two or more series
Facilitate the use of other
statistical measures.
Reliable
Determine the reliability of
an average
Why Measure of Dispersion
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.
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.
Types of Measures of
Dispersion
Classification of
Measures of dispersion
in Statistics
Measures of
Dispersion
Algebraic
Absolute Relative
Graphical
Algebraic Measure of Dispersion
× Mathematical way to calculate the
measure of dispersion.
Example: Calculation of Standard Deviation
or Co-efficient of Variance by using numbers
and formulas.
Characteristics of Algebraic
Measure of Dispersion
• Mathematical Way
• Algebraic Variables are used
• Numerical Figures are used here
• Formulas & Equations are used
Graphical Measure of Dispersion
× 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.
Characteristics of Graphical
Measure of Dispersion
• It is a visual way of measuring dispersion
• Graphs, figures are used
• Sometimes, it cannot give the actual result
• It helps the reader to have an idea about the
dispersion practically at a glance
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.
Classification of
Algebraic Measure of
Dispersion
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.
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
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
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
Absolute measures of Dispersion
Classification of Absolute measure
Mean Deviation
Quartile Deviation Standard Deviation
Range
“Range
Range
The difference between the maximum and
minimum observations in the data set.
R= H-L
5, 10 , 15 , 20, 7, 9, 12 , 17 , 13 , 6 , 10 , 11
, 17 , 16
Range = H- L
= 20- 5 = 15
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
“Mean deviation
Mean deviation
The average of the absolute values of
deviation from the mean(median or mode) is
called mean deviation.
=𝒇 | 𝒙−𝒙 |
𝑵
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
Demerits of Mean Deviation
Not reliable
Mathematically illogical to assume all
negatives as positives
Not suitable for comparing
series
“Quartile Deviation
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 – Q1
𝟐
Has better result than
range mode.
Is not affected by
extreme items
Merits of Quartile Deviation
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.
“Standard Deviation
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.
= √ 𝒙− 𝒙
𝟐
𝒏
68.2%
95.4%
99.7%
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.
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.
Classifications ofRelative Measures of
Dispersion
Chart of classification
Relative Measure
Coefficient of Range
Coefficient of Quartile
Deviation
Coefficient of Mean
Deviation
Coefficient of Variation
Coefficient of
Range
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= 𝑅𝑎𝑛𝑔𝑒
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑉𝑎𝑙𝑢𝑒+𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒× 100
“Coefficient of
Quartile Deviation
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 = 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑑𝑖𝑎𝑛× 100
= Q3 – Q1
Q3 + Q1
× 100 [By Simplification]
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
Coefficient of
Mean Deviation
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= 𝑀𝑒𝑎𝑛 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑑𝑖𝑎𝑛 𝑜𝑟 𝑀𝑒𝑎𝑛× 100
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
Coefficient of
Variation
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= 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑎𝑛× 100
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
Practical Uses of Coefficient of Variance
INVESTMENT ANALYSIS
STOCK MARKET
RISK EVALUATION
COMBINED STANDARD DEVIATION OF SEVERAL GROUPS
PERFORMANCES OF TWO PLAYERS
INDUSTRIES & FACTORIES
MathematicalApplication
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 = 𝐑
XH + XL × 100
= 𝟔
𝟕+𝟏× 100 =75%
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 𝐍+𝟏
𝟒th item = 130
And, Q3 = size of 𝟑(𝐍+𝟏)𝟒
th item = 215
So, Coefficient of Quartile Deviation =Q3 – Q1Q3 + Q1
× 100
= 215−130215+130 × 100= 24.64 %
Coefficient of Mean Deviation
Let the ages of 5 boys in a class is 12, 14, 14, 15, 18.So their Mean, 𝐱 = 𝟏𝟐+𝟏𝟒+𝟏𝟒+𝟏𝟓+𝟏𝟖
𝟓= 14.6
Mean Deviation, MD = | 𝒙 − 𝒙 |
𝑵
=|12−14.6| + |14−14.6| + |14− 14.6|+ |15−14.6| + |18−14.6|𝟓
= 1.52
Now, the Coefficient of MD= 𝐌𝐃
𝐱× 𝟏𝟎𝟎 = 𝟏.𝟓𝟐
𝟏𝟒.𝟔× 𝟏𝟎𝟎 = 10.41%
Coefficient of
Variation Suppose the returns on an investment for 4 years is Tk.1000, Tk.3000, Tk.4500 & Tk.5000.
So, Mean, 𝐱 = 3375 Standard Deviation, SD = 1796.99
So,Coefficient of Variation, CV= 𝐒𝐃
𝐱× 100
= 𝟏𝟕𝟗𝟔.𝟗𝟗𝟑𝟑𝟕𝟓
× 100 = 53.24%
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.
“
No of Days sale of sugarMonday 60Tuesday 120Wednesday 10Thursday 50Friday 70Saturday 140
𝜮 𝒐𝒇 𝑫𝒂𝒚𝒔 = 𝟔 𝜮𝒙 = 𝟒𝟓𝟎
Mean, 𝑥 = 𝑥
𝑛=
4𝟓𝟎
6= 7𝟓
“
x 𝒙𝟐
60 3600120 1440010 10050 250070 4900
140 19600𝜮𝒙 = 𝟒𝟓𝟎 𝜮𝒙𝟐= 45100
Standard deviation: 𝝈 =𝜮𝒙𝟐
𝒏−
𝜮𝒙
𝒏
𝟐=
𝟒𝟓𝟏𝟎𝟎
𝟔−
𝟒𝟓𝟎
𝟔
𝟐=
𝟕𝟓𝟏𝟔. 𝟔𝟔 − 𝟓𝟔𝟐𝟓 = 𝟒𝟑. 𝟒𝟗
Quartile Deviation
The marks of 7 students in Mathematics result are given below :
70, 85, 92,68, 75, 96, 84Find out-
• First Quartile Deviation• Third Quartile Deviation
Quartile deviation
× First quartile
𝐐𝟏 = 𝐬𝐢𝐳𝐞 𝐨𝐟𝐧 + 𝟏
𝟒
𝐭𝐡
𝐢𝐭𝐞𝐦
= size of 𝟕+𝟏
𝟒
𝐭𝐡𝐢𝐭𝐞𝐦
= size of 2nd item.= 70
×Third Quartile
𝑸𝟑 = 𝒔𝒊𝒛𝒆 𝒐𝒇𝟑 𝒏 + 𝟏 𝒕𝒉
𝟒𝒊𝒕𝒆𝒎
= size of 𝟑 𝟕+𝟏 𝒕𝒉
𝟒𝒊𝒕𝒆𝒎
= size of 6th item=92
Arranging the data in ascending order we get,68,70,75,84,85,92,96
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