![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?
1](https://image.slidesharecdn.com/session2-121004052554-phpapp02/85/Session-2-1-320.jpg)
![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)]
3](https://image.slidesharecdn.com/session2-121004052554-phpapp02/85/Session-2-3-320.jpg)
Session 2
- 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? 1
- 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. 2
- 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)] 3