Measure of dispersion

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Assignment
Submitted to: SIRRAFIQUEDAUDPOTO
NAME: WAQAR MEHMOOD KALHORO
ROLL: 2K19/AA/43
Department of ANTHROPOLOGY&ARCHAEOLOGY
Topics: Measure of Dispersion

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Topic
No;
INDEX Page No;
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MEASURE
CLASSIFICATION OF MEASURES OF DISPERSION
RANGE
RANGE OF SERIES
QUARTILE DEVIATION
MEAN DEVIATION
STANDARD DEVIATION - S
RELATIVE MEASURE OF DISPERSIONS
CO-EFFICIENT OF RANGE
CO-EFFICIENT OF MEAN DEVIATION
CO-EFFICIENT OF QURATILE DEVIATION
DIFFERENCE BETWEEN ABSOLUTE AND RELATIVE
MEASURE OF DISPERSION
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12
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bi-smillāhir-raḥ
māni r-raḥ
īmⁱ, "In the nameofGod, the MostGracious,theMost Merciful."

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Measure of Dispersion
Definition; A measure of dispersion, computed by talking arithmetic mean of
absolute value of the devition of the functional value form the central value usually
the mean or mode.
→The mean of distance of each value form their mean.
→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.
Dispersion is contrasted with location or central tendency, and together they are the
most used properties of distributions.
Measures
A measure of statistical dispersion is a nonnegative real number that is zero if all
the data are the same and increases as the data become more diverse.
Most measures of dispersion have the same units as the quantity being measured.
In other words, if the measurements are in metres or seconds,so is the measure of
dispersion. Examples of dispersion measures include:
Standard deviation
Interquartile range (IQR)
Range
Mean absolute difference (also known as Gini mean absolute difference)
Median absolute deviation (MAD)
Average absolute deviation (or simply called average deviation)
Distance standard deviation
These are frequently used (together with scale factors) as estimators of scale
parameters, in which capacity they are called estimates of scale. Robustmeasures
of scale are those unaffected by a small number of outliers, and include the IQR
and MAD.

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All the above measures of statistical dispersion have the useful property that they
are location-invariant and linear in scale. This means that if a random variable X
has a dispersion of SX then a linear transformation Y = aX + b for real a and b
should have dispersion SY = |a|SX, where |a| is the absolute value of a, that is,
ignores a preceding negative sign –.
Other measures of dispersion are dimensionless. In other words, they have no units
even if the variable itself has units. These include:
Coefficient of variation
Quartile coefficient of dispersion
Relative mean difference, equal to twice the Gini coefficient
Entropy: While the entropy of a discrete variable is location-invariant and scale-
independent, and therefore not a measure of dispersion in the above sense, the
entropy of a continuous variable is location invariant and additive in scale: If Hz is
the entropy of continuous variable z and y=ax+b, then Hy=Hx+log(a).
There are other measures of dispersion:
Variance (the square of the standard deviation) – location-invariant but not linear
in scale.
Variance-to-mean ratio – mostly used for count data when the term coefficient of
dispersion is used and when this ratio is dimensionless, as count data are
themselves dimensionless, not otherwise.
Some measures of dispersion have specialized purposes, among them the Allan
variance and the Hadamard variance.
For categorical variables, it is less common to measure dispersion by a single
number; see qualitative variation. One measure that does so is the discrete entropy.

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Classification of Measures of Dispersion
RANGE
 Range is defined as the difference between the maximum and the minimum
observation of teh given data.
 If Xm the miximum observation ,
X0 the minimum observation,
then
Range = Xm-X0
RANGE OF SERIES
 Individual Series
In case of individual series, the difference between largest value and smallest
vale can be determined and it is called range.
 Discrete Series
To find the range; first` order the data form least to greatest. Then subtract the
smallest value from the largest value in the set.
 Continuous Series
Relative
Co-Efficient
of Range
Co-Efficient
of Standaed
Deviation
Co-Efficient of
Quratile
Deviation
Co-
Efficient of
Mean
Deviation
Absolute
Range
Standard
Deviation
Quartile
Deviati
on
Mean
Deviati
on

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 In case of continous frequency distribution, range, according to the
definition, is calculated as the differece between the lower limit of the
minimum interval and upper limit of the maximum interval of the grouped
data.
 Example,
Range of following series is 40-0=40.
Class Boundaries Frequency
0-10 12
10-20 8
20-30 10
30-40 5
40-50 7
QUARTILE DEVIATION
“One half of the inter quratile range is called quartile deviation”
 A simple way to estimate the spread of a distribution about a measure of its
central tendency.
 The difference Q3-Q1 is called the inter quartile range.
IQR= Q3- Q1 Qd=Q3 – Q1
2
QUARTILES
 Quratiles are used to divided a given dataset into four equal halves.

Q1
25%
Q2
50%
Q3
75%
Q4
100%

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 The first quartil or the lower quratile is the 25th percentil, also denoted
by Q1.
Q1= N+ 1th
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 The thrid quartile or the upper quartial is the 75th percentil, also denoted
by Q3.
Q3=3.(N+1)
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Shoted Data-5,10,15,17,18,19,20,21,25,28 n(number of data) = 10
First Quartile Q1 = (n+1/4) th term
=10+1/4th term =2.75th term
= 2nd term =0.75*(3rd term- 2nd term) =10+0.75*(15-10)
= 10+3.75=13.75
Third Quartile Q3 = 3(n+1/4)th term.
=3(10+1) 4th term = 8.25th term
=8th term +0.25*(9th term- 8th term) = 21+0.25*(25-21)
=21+1=22
 Quratile Deviation = semi-Inter Quratile Range
= Q3 – Q3 * 2
=22-13.752*2
=8.252*2
=4.125
MEAN DEVIATION
 The average of the absolute value of deviation from the mean is Mean
deviation.
 FORMULA;

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Formula of mean deviation is,
 For sample data
Mean Deviation=∑(x-x)
n
 For frequency distribution
M.D= ∑f(x-x
̅ )
∑f
Where;
o X= Each value or observation
o X
̅ = Mean
o n= Number of value
Example
 Set of value is (1,2,3,4,5)
X
̄ is Mean =(15÷5)=3
 The difference between this X
̄ and the value in the set is
(2,1,0,-1,-2) and sum of values =6
Mean Deviation =(6÷5)=1.2
VARIACE – α2
/ s2
 The variance is the average of the squared difference between each data
value and the mean.
 The variance is computed as follows:
Sample variance = s2 = ∑(X-X
̄ )2
n-1
STANDARD DEVIATION - S
 Standard deviation is calculated as the square root of average of squared
deviation taken from actual mean.

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 It is also called Rootmean square deviation.
√∑(𝑥 − X̄)2
n-1
MERITS OF STANDARD DEVIATIONS
 This measure is most suitable for making comparsions among two or more
series about variability.
 It takes into account all the items and is capable of future algebric treatment
and statistical analysis.
DEMERITS OF STANDARD DEVITIONS
 It is difficult to complete
 It assigns more weight to extreme item and less weight to items that are
nearer to mean.
RELATIVE MEASURE OF DISPERSIONS
Co-Efficientof Range
“The relative measure of the distribution based on range is known as the
coefficient range”.
Where
 The difference between the miximum and minimum values of a given set of
data know as the range,
Formula
Coefficient of Range = (xm – x0)
Where
Xm =Maximum Value
X0 = Minimum Value
EXAMPLE
Data set= 8,5,6,7,3,2,4

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Step1: Find Range
Range=Maximum Value- Minimum Value
Step2: Find Range Coefficient
Coefficient of Range =(Maximum Value- Minimum Value) /(Maximum Value
+Minimum Value)
Co-efficient of mean deviation
It is defined as ratio of mean deviation to the average used in the calculation of
mean deviation.
For sample data; Co-efficient of mean deviation =M.D
X
̅
For Population data;
Co-efficient of mean deviation = M.D
μ
Example;
 3, 6,6,7,8,11,15,16 find the mean deviation.
Solution;
Step#1 find the simple mean or Average
Mean=3, 6, 6, 7, 8, 11, 15, 16/9
Mean=72/9
Mean= 7
Steps:
 Three steps to find mean,
I. Find the mean of all values
II. Find the difference of each value from that mean
III. Find the mean of those differences.
Example: 271,354,296,301,331,320,285,298,327,316,287,314
Find the mean deviation
Solution#
Step1# finds the simple mean,
=271+354+296+301+331+320+285+298+327+316+287+314
12
=3700
12
= x
̅ =Mean=308

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Step2# Find the absolute value from that mean
X X-X
̅ Absolute Value
271 271-308 37
354 354-308 44
296 296-308 12
301 301-308 7
333 333-308 23
326 326-308 16
285 285-308 23
298 298-308 10
327 327-308 19
287 287-308 21
316 316-308 8
314 314-308 6
∑(x-x
̅ )=226
Step#3
Find the mean deviation
M.D =∑(x-x
̅ )
n
= 226
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M.D = 18.83
Co-efficient of mean deviation = 18.83 = 0.061
308
COEFFICIENT OF VARIANCE
 Also known as relative standard deviation (RSD)
 It is defined as the ratio of standard deviation to mean.
 Formula
CV= s/µ
Where
s = standard deviation
µ =mean

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EXAMPLE
 The coefficient of variation can also be used to compare variability between
different measures.
MERITS OF CV
 Widely used in analytical chemistry to express the precision and
repeatability of an experiment.
 Used in fields such as engineering or physics when doing quality assurance
studies,
 Utilized by economists and investors in economic models.
CO-EFFICIENT OF QURATILE DEVIATION
 A relative measure of dispersion based on the quartile deviation is called the
coefficient of quartile deviation.
 Also called quartile coefficient of dispersion.
Coefficient of
=Q3– Q1 x 100
Quartile Deviation Q3 – Q1
From EXAMPLE OF QUARTILE DEVIATION
 First Quartile Q1=13.75
 Third Quartile Q3 = 22
 Coefficient of QV =22–13.752 = 8.25 = 0.23*100=23
22+13.75 35.75
Regular Test Randomized Answers
SD 10.2 12.7
Mean 59.9 44.8
CV% 17.3 28.35

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DIFFERENCE BETWEEN ABSOLUTE AND RELATIVE MEASURE OF
DISPERSION.
ABSOLUTE MEASURES
 An absolute measure is one
that uses numerical variations
to determine the degree of
error.
 Measure the extent of
dispersion of the item values
from a measure of central
tendency.
 They are expressed in terms of
the original units of the series.
 Useful for understanding the
dispersion within the context
of experiment and
measurements .
 Comparatively easy to
compute and comprehend.
RELATIVE MEASURE
 Use statistical variations based
on percentages to determine
how far from reality a figure is
within context.
 Are known as ‘Coefficient of
dispersion’- obtained as ratios
or percentages.
 They are pure numbers
independent of the unite of
measurement.
 Useful for making
comparisons between
separated data sets or different
experiments.
 Comparativerly difficult to
compute and comprehend.

Measure of dispersion

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