Download Introduction - GCG-42

Introduction
Meaning
Dispersion
• Central Tendency alone does not explain the observations
fully as it does reveal the degree of spread or variability of
individual observations in a series. Measures of dispersion
help us understanding the variability of items.
The term dispersion is used in two senses1. Firstly dispersion refers to variation of items amongst
themselves. e.g. if the value of all the items are same, it
will have zero dispersion.
2. Secondly, dispersion refers to variation of items around
an average. If the difference between the value of items
and average is large, dispersion will be high and if the
difference is small, dispersion will be low.
“Dispersion is a measure of variation of items.”-Bowley
Definitions
“The degree to which numerical data tend to spread about
average is called variation or dispersion of data.”-Spiegel
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Objectives
Dispersion
1. To determine the reliability of average- Dispersion helps in
determining reliability of average by pointing how far is single
average figure is representative of data. If the dispersion is small,
the average is a reliable indicator of data.
2. To compare the variability of two or more series- It helps
comparison of two or more series of data. The high degree of
variability means less consistent data.
3. Facilitates use of other statistical measures-Dispersion serves
the basis of other statistical measures like correlation, regression,
testing of hypothesis etc.
4. Statistical quality control– Identifies whether the variation in
quality of a product is due to random factors or is there some
other defect in the manufacturing process.
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Dispersion
Properties & types
Good
Average
Properties of good dispersion are1. It should be simple to compute.
2. Should be easy to understand.
3. Should be uniquely defined.
4. Should be based on all observations without unduly
affected by extreme observations.
5. Should be capable for further algebraic treatment.
Dispersion
Absolute or
Relative
Absolute
Measure of
dispersion
expressed in the
same unit in which
data of series is
given
Relative
Measure of dispersion
expressed in the
percentage or ratio. It is
also called coefficient of
dispersion.
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Methods of Measurement
Dispersion
There are 3 main methods of dispersion1. Range
2. Interquartile range and quartile deviation
3. Mean Deviation
Definition
Individual
series
Range-It is defined by difference between maximum and
minimum value of a series or a data set.
Coefficient of Range- It is relative measure of dispersion and
is also called range coefficient of dispersion.
Coefficient of range= (xmax-xmin)/(xmax+xmin)
Interquartile Range- The difference between the upper
quartile (Q3) and the lower quartile (Q1) is called
interquartile range.
Quartile Deviation-It is half the difference between upper and
lower quartile i.e. (Q3-Q1)/2.
The relative measure of quartile deviation is called coefficient
of quartile deviation and is defined as=(Q3-Q1)/(Q3+Q1)
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Mean Deviation
Mean
Deviation
Dispersion
Mean deviation is another measure of dispersion and is also
known as average deviation. It is defined as arithmatic
average of deviation of various items of a series computed
from some measures of central tendency say median or
mean. However median is preferred because the sum of
deviations of item taken from Median is minimum when signs
are ignored. The formulae for calculating mean deviation are-
For calculating coefficient of mean deviation-
Coefficient
of mean
deviation
For continuous series, the mid points of various classes and
deviations from these values are used to calculate mean
deviation and coefficient of mean deviation.
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Standard Deviation
Dispersion
It is most widely used measure of dispersion and is also
called root mean square deviation. It is calculated as square
root of arithmatic mean of the squares of the deviation of the
values taken from the mean. It is calculated by-
Where X is mean value with mean value µ meaning
E[X]= µ where as
is sigma (standard deviation)
Introduction
Coefficient of SD- Coefficient of SD (relative measure) is
obtained by dividing standard deviation by the arithmatic
average. The formula is
__
Where X is arithmatic average.
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Mean Deviation vs SD
Differences
Calculation
of SD
Dispersion
Both are measures of dispersion but they are different in some
ways
1. Algebraic signs of deviations are ignored while calculating
mean deviation where as in the calculation of standard
deviation, the signs of deviations are taken into account.
2. Mean deviation can be computed from either of mean, median
or mode where as SD is always computed from mean because
sum of squares of deviations taken from mean is minimum.
In case of individual series, SD can be calculated by
1. Actual mean method
2. Assumed mean method (in case actual mean is not whole no.)
3. Method based on actual data (when observations are less)
In case of discrete series SD can be calculated by
1. Actual mean method
2. Assumed mean method (or short cut method)
3. Step deviation method (used to simplify calculations by dividing
deviations by a common factor)
In case of continuous series- Under this all the methods used in
discrete series can be used as classes are represented by mid
values.
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Dispersion
Variance
Introduction
Variance is another measure of dispersion. Variance is square of
Standard Deviation.
Variance= (SD)2
Variance is calculated by 3 methods as under
The standard deviation of two groups can be calculated by
Combined
standard
deviation
Where
12=Combines
standard deviation
1 =SD of first group
2= SD of second group
d1=X1-X12
d2=X2-X12
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