Professor Vipin 2015
Measures of Dispersion
Absolute Measures of Variation
1.
2.
3.
4.
Range
Quartile Deviation
Mean Deviation
Standard Deviation
Relative Measures of Variation
1.
2.
3.
4.
Coefficient of Range
Coefficient of Quartile deviation
Coefficient of Mean deviation
Coefficient of Standard deviation
Range
It is the simplest method of studying variation.
Coefficient of Range
Quartile Deviation
It is the half of the inter quartile range.
Coefficient of QD
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Professor Vipin 2015
Mean Deviation (From Mean)
1. Ungrouped Data
∑|
̅|
2. Discrete Data (Grouped)
∑ |
̅|
3. Continuous (Grouped) is a relative measure
( ̅)
̅
Mean Deviation (From Median)
1. Ungrouped data
∑|
|
2. Discrete data (Grouped)
∑ |
|
3. Continuous (Grouped)
( )
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Professor Vipin 2015
Mean Deviation (From Mode)
1. Ungrouped Data
( )
∑|
|
2. Discrete data (Grouped)
∑ |
|
3. Continuous (Grouped)
It is the same as discrete data. However x denotes the mid points of the CI
Standard Deviation
It is the positive square root of the mean of the squared deviations of given observations from their
AM.
1. Ungrouped Data
Direct Method
∑(
√
̅)
∑
√
(
∑
√
(
∑
)
Deviation Method
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∑
)
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Professor Vipin 2015
2. Discrete Data (Grouped)
∑
√
(
∑
√
(
∑
√
(
∑
)
Deviation Method
∑
)
3. Continuous Frequency (Grouped)
∑
)
Where x is the midpoint of the class interval
Deviation Method
∑ ( ́)
√
(
∑
́
)
́
Coefficient of SD
̅
Coefficient of Variation
̅
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Professor Vipin 2015
Moments
Meaning
Moments are used to describe characteristics of a frequency distribution such as averages,
dispersion, skewness and kurtosis.
Types of Moments
1. Raw Moments
Moments about any arbitrary point ‘A’ are known as raw moments.
∑(
)
2. Central Moments
Moments about the mean are known are central moments.
a) First
∑(
x)
b) Second
∑(
x)
c) Third
∑(
x)
∑(
x)
d) Fourth
Karl Pearson’s Beta and Gamma Coefficients
These are coefficients based on the 1st four central moments.
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Professor Vipin 2015
Also
√
Tells us if a distribution is skewed or whether it is symmetrical and
between a symmetrical curve and normal curve.
tells us the difference
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability
distribution of a real-valued random variable about its mean. The skewness value can be positive or
negative, or even undefined.
Types of Skewness
Teacher expects most of the students get good marks. If it happens, then the cure looks like the
normal curve below:
But for some reasons (e. g., lazy students, not understanding the lectures, not attentive etc.) it is not
happening. So we get another two curves.
Karl Pearson’s Coefficient of Skewness
x
Bowley’s Coefficient of Skewness
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