Measures of Dispersion

It has two terms:
□ Measure: It means a “specific method of estimation”
□ Dispersion: (also known as scatter, spread, variation) the term means
“ difference or deviation of a certain values from their central
value”

“ The measurement of the degree of
variation or the extent to which items vary
from their central value in a population or
sample”

● To compare the variability of two or more data set.
● To serve as the basis for the control of variability.
● To determine the reliability of an average.
● To facilitate the use of other statistical measures.

◦ It should be rigidly defined.
◦ It should be easy to understand & calculate.
◦ It should be based on all observations of a data.
◦ It should be easily subjected for further mathematical operations.
◦ It must be least affected by the sampling fluctuation.

Absolute
Range
Standard
Deviation
Quartile
Deviation
Mean
Deviation Relative
COEFFICIENT
OF RANGE
COEFFICIENT
OF
STANDARD
DEVIATION
COEFFICIENT
OF QUARTILE
DEVIATION
COEFFICIENT
OF MEAN
DEVIATION
Classification of measures of
Dispersion

 Range is defined as the difference between the maximum and the
minimum observation of the given data.
 If Xm the maximum observation,
X0 the minimum observation
then
Range = X m – X0

 Individual Series
In case of individual Series, the difference between largest value and
smallest value can be determined and it is called range.
 Discrete series
To find the range; first’ order the data from least to greatest. Then
subtract the smallest value from the largest value in the set.

CONTINUOUS SERIES
 In case of continuous frequency distribution, range, according to
the definition, is calculated as the difference 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

“ One half of the inter quartile 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.

 Quartiles are used to divide a given dataset into four equal halves.
Q1
25%
Q2
50%
Q3
75%
Q4
100%

 The first quartile or the lower quartile
is the 25th percentile, also denoted by Q1.
 The third quartile or the upper quartile
is the 75th percentile, also denoted by Q3.

Sorted 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

 Quartile Deviation = Semi-Inter Quartile Range
= Q3–Q1× 2
= 22–13.752× 2
= 8.252 × 2
= 4.125

 The average of the absolute values of deviation from the mean is called
mean deviation.
 Formula
M.D from mean = ∑ ∣X−mean∣ /n
Where,
X = Given values
n = Total no. of values

EXAMPLE
 Set of values is ( 1, 2, 3, 4,5 )
x̅ is Mean = (15 ÷ 5) = 3
 The difference between this x̅ and the values in the set is
(2, 1, 0, -1,-2) and sum of set values = 6
Mean Deviation = (6 ÷ 5) = 1.2

𝜎2 /𝑠2
 The variance is the average of the squared difference between each
data value and the mean.
 The variance is computed as follows :

S
 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 .

 This measure is most suitable for making comparisons among two
or more series about variability .
 It takes into account all the items and is capable of future algebraic
treatment and statistical analysis .

It is difficult to
complete It assigns more
weights to extreme
item and less
weights to items
that are nearer to
mean.

“The relative measure of the distribution based on range is
known as the coefficient range.’’
Where,
• The difference between the maximum and minimum values of
a given set of data known as the range.

FORMULA
Coefficient of Range = (xm- xo)/(xm + xo)
Where,
xm = Maximum Value
xo = Minimum Value

EXAMPLE
Data set = 8, 5, 6, 7, 3, 2, 4
Step 1: Find Range
Range = Maximum Value - Minimum Value
Step 2: Find Range Coefficient
Coefficient of Range = (Maximum Value - Minimum Value) / (Maximum
Value + Minimum Value)

A relative measure of dispersion based on the mean deviation is called the
coefficient of the mean deviation or the coefficient of dispersion.
Coefficient of M.D. = Mean Deviation about A *100
A
Where,
A can be mean,mode or median

 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

EXAMPLE
 The coefficient of variation can also be used to compare variability between different
measures.
Regular Test Randomized Answers
SD 10.2 12.7
Mean 59.9 44.8
CV % 17.03 28.35

 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

 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
Quartile Deviation
Q3–Q1
Q3+Q1
= ×100

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

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.
Relative measures
 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.

Absolute measures
 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 measures
 They are pure numbers
independent of the units of
measurement.
 useful for making comparisons
between separate data sets or
different experiments
 Comparatively difficult to
compute and comprehend

Measures of Dispersion

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