The document discusses measures of dispersion such as variance, standard deviation, and the coefficient of variation. It defines variance as the average squared deviation from the mean and standard deviation as the positive square root of the variance. The coefficient of variation measures relative dispersion by dividing the standard deviation by the mean. It is unit-free and allows for comparison across distributions. The document also covers Chebyshev's inequality and how it relates to the proportion of data within a given number of standard deviations from the mean.

1. Measures of Dispersion Variance and the standard deviation
• Concept and objective Variance is the average squared deviation from the mean
• Range, inter-quartile range (IQR =Q3 – Q1)
• Variance and standard deviation (sd) 1 N
• Computation
σ2 =
N
∑(X
i =1
i − µ )2
x x x x
• Chebyshev’s Inequality X1 Xi µ X2 XN
• Relative dispersion -- the coefficient of
σ, the standard deviation is the (+ve) square root of the variance
variation
In the sample variance calculation, use the denominator (n-1)
σ
C.V. = × 100%
µ 1 n
S2 = ∑ ( X i − X )2
n − 1 i =1
Chebyshev’s Inequality
Chebyshev’s Inequality:
• At least (1 - 1/k2) proportion of the data must be
within k standard deviation of the mean. Illustration
• Here k is a number (not necessarily an integer)
greater than 1.
• The statement is valid for ANY distribution,
discrete or continuous, symmetric or otherwise.
• If a r.v. X has a mean µ and a standard deviation σ
then a equivalent probability statement is :
µ-3σ µ-2σ µ-σ µ µ+σ µ+2σ µ+3σ
1
P [ | X − µ | > kσ ] ≤ At least 75%
k2
At least 88.89%
Understanding Standard deviation
The Coefficient of Variation:
294 MLA of WB has an average wealth of 68 Lakh A measure of relative dispersion
and s.d. = 10 Lakh. What does that tell you? σ S
or × 100 %
In particular, what can you say about % of MLAs µ X
having wealth • Unit free
• between 58L and 78L ? • Amenable to comparison
• Often expressed in terms of percentages
• between 53L and 83L ?
• Less than 53 L or more than 83L?
• between 50L and 1 crore ?
• More than 1 crore?
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2. Box Plot Box Plot
Elements of a Box Plot
Smallest data Largest data point
point not below not exceeding Suspected
Outlier inner fence inner fence outlier
o X X * o X X
*
Inner Q1 Median
Outer Q3 Inner Outer
Fence Fence Fence Fence
Q1-1.5(IQR) Interquartile Q3+1.5(IQR)
Range
Q1-3(IQR) Q3+3(IQR)
Review Descriptive Statistics
Poll Forecasting – Exit polls
• Graphical representations, frequency
distribution • Exit Polls in US election 2000 in the critical
state of Florida
• Measures of central tendency and dispersion
– Computation and interpretation • Indian Election 2004, 2009,
– Chebychev’s result • UP 2012
• Skewness and Kurtosis • Karnataka 2008
• Outliers
– What are they? How to detect?
– What to do if there are outliers?
A simple example: Overview Some of the questions to be
Population = all projects undertaken by a company answered
An unknown proportion π of them took longer than scheduled
* • To what degree, the randomness in Y can be
* Delay *
* ** * * attributed to sampling fluctuations?
* *
*
* *
• How close is Y = p to π ?
* * n
**
* * * * * • If we want p to be within ±0.05 of π , how
many projects do we need to look at?
A random sample of n projects are selected. Y of them are found to be delayed.
(sample outcome) Y is random, but the randomness depends on π.
Given the value of Y, one can make an objective inference about π.
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3. Myth and Mystery of Probability
Overview of Probability
• What is chance of getting any rain today in • Approaches for defining probability
the campus?
• What is the probability that India will win • Basic Probability rules
WC2015?
• What is the chance that India’s space • Conditional probability and notion of
mission will send a human being to moon independence
by 2020?
• Bayes’ rule
Approaches for defining
Probability Laws
probability
• Classical approach • 0 ≤ P[A] ≤1;
• P[impossible event]=0; P[Sure event]=1
• (Asymptotic) Relative frequency approach • P[A or B] = P[A] + P[B] - P[AB]
• In particular, P[not A] = 1- P[A]
• Subjective probability • Look at the Venn diagram and write down
other formulae like
• P[A] = P[A and B] + P[A and (not B)]
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