Measure of Dispersion
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Why to measure dispersion?
• An average can represent a data-set only as best as a single value can, but
it certainly cannot reveal the entire story of the data-set. For example,
consider the following data sets:
i) 5, 5, 5, 5, 5; (mean = 5, median 5)
ii) 3, 4, 5, 6, 7: (mean = 5, median =5)
iii) 1, 3,5, 7, 9: (mean = 5. median =5).
• In all three data-sets above, the average values (mean and median) are the
same.
• What we observe here is, an average fails to give any idea about the spread
or scatteredness of the observations in the data-set.
• This scatteredness is also called dispersion.
• Some important measures of dispersion are range, mean deviation, and
standard deviation
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Introduction
• ‘Measures of dispersion’ measures the degree or extent of scatterness (or
spread or variation) of items about the central value data.
• It measures the way into which the data in distributed and thus facilitates to
compare two or more distributions to get a precise information about them.
• Dispersion can also be called as variability, scatter or spread.
• It helps to interpret the variation of the data from one another that is to know
how much homogenous or heterogeneous the data is, and gives a clear
picture about the distribution of the data.
• The main goal of measure of dispersion is to get to know how the data is
scattered.
• It illustrates the variation in data from their average value.
• In simple wards, it shows how squeezed or scattered the variable is. Thus,
dispersion is the extent to which a distribution is stretched or squeezed.
• Dispersion is different than the location or central tendency. Dispersion with
central tendency is the most used property of distributions.
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Measures of dispersion are needed basically
(i) to determine the reliability of data.
(ii) as it serves as a basis to control the variability.
(iii) to compare two or more series with respect to variability.
(iv) many statistical tools are based on it.
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Types of Measures of dispersion
Absolute measure of dispersion Relative measure of dispersion
• obtained as ratios or percentages of the
• contains the same unit as the original average.
data set. • also known as coefficient of dispersion.
• expressed in the unit of variable itself • used to evaluate the distribution of two or
like kilograms, rupees, centimeters, more data sets.
marks etc. • data values are compared without their units.
• states the variations as average of These are pure numbers or percentages totally
deviations of observations like standard independent of the units of measurements.
or means deviations. • e.g. coefficients of range, variation, SD.
• E.g. Range, SD, quartile deviation, etc. quartile deviation and mean deviation,
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respectively.
Various Measures of Dispersion
• The various measures of dispersion are:
1. The range
2. Quartile and semi-quartile range or interquartile range
3. Mean Deviation
4. Variance and Standard deviation (S.D.) and
5. Coefficient of variation (C.V.)
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1. The Range
Range is defined as the difference between two extremes observations
of the data set such as lowest and highest value.
It is the simplest measure of dispersion.
Symbolically,
Range = L-S
Where L = Largest item
S = Smallest item
The relative measure of range is called the coefficient of range and is
given by
𝐋−𝐒
Coefficient of range = www.DuloMix.com 7
𝐋+𝐒
• In case of continuous series the range can be found out by
(i) finding the difference between the upper limit of the highest class
and the lower limit of the lowest class.
(ii) finding out the difference between the mid-point of the highest
class interval and mid-point of the lowest class internal.
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Merits of Range:
1. It is the simplest form of the measure of dispersion
2. It is easy to calculate and to understand.
3. It is independent of change of origin.
Demerits of Range:
1. It is based on two extreme observations, hence get affected by
fluctuations.
2. Although range is simplest measure of dispersion it is not reliable.
3. It is dependent on change of scale.
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Exercise: The percentage of marks obtained by six students in English
are 20, 25, 80, 30, 90 and 45. Calculate the range and coefficient of
range.
Solution: Here L = 90 and S 20
Range = L- S = 90 – 20 = 70% mark
𝐋−𝐒 𝟗𝟎−𝟐𝟎 𝟕𝟎
Coefficient of Range = = =
𝐋+𝐒 𝟗𝟎+𝟐𝟎 𝟏𝟏𝟎
= 0.64
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Exercise: Find the range and coefficient of range from the data given
below:
Weights (lbs) 85-95 95-105 105-115 115-125 125-135
No. of persons 4 8 12 14 7
Solution :
Here, L= 135 and S= 85
Range = L –S = 135-85
= 50 lbs
𝐋−𝐒
Coefficient of range=
𝐋+𝐒
𝟏𝟑𝟓−𝟖𝟓 𝟓𝟎
= =
𝟏𝟑𝟓+𝟖𝟓 𝟐𝟐𝟎
= 0.227
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Example: Age of sample of 10 patients from a population of 169
patients is given in Table. Determine range and coefficient of range.
Patient No. 1 2 3 4 5 6 7 8 9 10
Age (Years) 42 28 28 61 31 23 50 34 32 37
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Standard Deviation
• Standard deviation is most widely used measure of dispersion of a
series and is commonly denoted by the symbol ‘𝜎′(pronounced as
sigma).
• Standard deviation is defined as the square-root of the average of
squares of deviations, when such deviations for the values of
individual items in a series are obtained from the arithmetic average.
• The square od SD is called variance.
• So SD = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
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• SD is a measure of the amount of variation or how spread out
numbers in a set of values.
• The low value of SD indicates that the values tend to be close to the
mean, means high degree of uniformity also called the ‘expected
value’ of the data set.
• The high value of SD indicates that the values are spread out over a
wider range.
• The SDs of a random variable, statistical population, data set and
probability distribution is the square root of its variance.
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For the ungrouped data x1, x2, x3 ,…… xn, the standard deviation is
1
s = Ʃ 𝑥 − 𝑥ҧ 2
𝑛
This formula can be simplified as
Ʃ𝑥2 Ʃ𝑥 2
s = −
𝑛 𝑛
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Example: The reactions (in mm) of tuberculin test of 10 boys are as follows:
8,3,7,5,8,11,10,9,7,12 Find the standard deviation of the data.
Ʃ𝑥 80
Solution : Mean 𝑥ҧ = = = 8 mm
𝑛 10
𝒙 𝒙 − 𝒙ഥ (𝒙𝒊 − 𝒙ഥ)2 1
s = Ʃ 𝑥 − 𝑥ҧ 2
𝑛 𝑖
8 0 0
3 -5 25
66
7 -1 1 =
10
5 -3 9
8 0 0
= 6.6 = 2.57mm
11 3 9
10 2 4
9 1 1
7 -1 1
12 4 16
total 66
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Another method
Total
𝒙 8 3 7 5 8 11 10 9 7 12 80
𝒙2 64 9 49 25 64 121 100 81 49 144 706
Ʃ𝑥2 Ʃ𝑥 2
s = −
𝑛 𝑛
706 80 2
= −
10 10
= 70.6 − 64
= 6.6
= 2.57
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Example: Calculate variance and S.D. of the following series.
18,20,22,27,21,29,27,29,28,29
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• When deviations are taken from assumed mean the fooling formula is
applied
Ʃ𝑥2 Ʃ𝑥 2 Ʃ𝑑2 Ʃ𝑑 2
s = − = s = −
𝑛 𝑛 𝑛 𝑛
Where , X-A = d And A = Assumed mean
Ʃ𝑓𝑥2 Ʃ𝑓𝑥 2 Ʃ𝑓𝑑2 Ʃ𝑓𝑑
s = − 2
= − × 𝑐
𝑛 𝑛 𝑛 𝑛
𝑥−𝐴
Where, d = , c= width of the class
𝑐
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Example : Compute the arithmetic mean and the standard deviations
for the following data.
Scores 4-5 6-7 8-9 10-11 12-13 14-15
Frequency 4 10 20 15 8 3
Solution: we construct the following table taking A= 8.5 and c =2
Class Frequency f Mid-value x 𝒙 −𝑨
d = fd fd2 Ʃ𝑓𝑑2 Ʃ𝑓𝑑 2
𝒄 S = − × c
𝑛 𝑛
4-5 4 4.5 -2 -8 16
6-7 10 6.5 -1 -10 10 100 22 2
= − × 2
8-9 20 8.5 0 0 0 60 60
10-11 15 10.5 1 15 15 = 1.667 − 0.1344 × 2
12-13 8 12.5 2 16 32
14-15 3 14.5 3 9 27 = 1.5323 × 2
Total 60 22 100 = 1.24 × 2
= 2.48
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Example: Find the standard deviation of the following distribution.
Age 20-25 25-30 30-35 35-40 40-45 45-50
No.of persons 170 110 80 45 40 35
𝒙 −𝑨
Solution : A =32.5 , c = 5 and d =
𝒄
Class Frequency f Mid-value x 𝒙 −𝑨
d = fd fd2 Ʃ𝑓𝑑2 Ʃ𝑓𝑑 2
𝒄 S = − × c
𝑛 𝑛
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Example: Calculate the mean and standard deviation of the following
table giving the age distribution of 542 members.
Age in years 20-30 30-40 40-50 50-60 60-70 70-80 80-90
No.of 3 61 132 153 140 51 2
members
𝒙 −𝑨
Solution : A = 55 , c = 10 and d =
𝒄
Class Frequency f Mid-value x 𝒙 −𝑨
d = fd fd2 Ʃ𝑓𝑑2 Ʃ𝑓𝑑 2
𝒄 S = − × c
𝑛 𝑛
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Example : A sample of 20 persons drawn from the same universe showed the
number of attacks of cold per person as follows:
7,2,5,0,1,5,5,6,7,6,2,6,9,4,8,8,6,7,9,3
Calculate mean , Standard deviation and standard error of the mean for above
data and comment on your results.
Solution: Let x denote the number of attacks of cold per person.
For the given data Ʃx = 106 and Ʃx2 = 690
Ʃ𝑥 106
Mean 𝑥ҧ = = = 5.3
𝑛 20
Ʃ𝑥2 Ʃ𝑥 𝑠
Standard deviation, s = − 2 Standard error of the mean =
𝑛 𝑛 𝑛
690 106
= − 2
2.53
20 20 =
20
= 34.5 − 28.09
= 6.41 = 0.5561
s = 2.53