statistics

Introduction to Probability and Statistics

Nagarajan Krishnamurthy

Introduction to Business Statistics for EPGP 2015-16 batchIndian Institute of Management Indore

Thanks to Prof. Arun Kumar and Prof. Ravindra Gokhale,co-instructors of QT1, AY 2012-13

Part 1: Summarizing and Visualizing a Data Set

Types of Data

Quantitative Data: Data for which arithmetic operationsmakes sense. E.g.: Age, Salary, Length.

Categorical Data: Data obtained by putting individuals indifferent categories. E.g.: Gender, States of a country

Visualization

Quantitative Data: Histogram, Stem-Leaf plot, Box plot

Categorical Data: Pie Chart, Bar chart

*Discuss Cafe data (using Excel)

Interpreting a Histogram

Shape: symmetric, skewed;unimodal, bimodal, ...;leptokurtic, platykurtic, mesokurtic

Center: mean, median

Spread: range, standard deviation, inter-quartile range

Measure of the central tendency of a data set

Mean: If we have a data set x1, . . . , xn then mean of the dataset is x1++xn

n.

Notation: x

Mean: Example

The mean of 0,5,1,1,3 is 2.

Measure of the Central Tendency of a Data Set

Median: Middle number in a sorted data set. When thenumber of observations (sample size) is an even number thenthere are two middle numbers. In that case, we take averageof the two middle numbers to obtain the median.

Notations: x

Median: Example 1

For example the median of 0,5,1,1,3 is 1 because 1 is themiddle number of the sorted data i.e. 0,1,1,3,5.

Median: Example 2

The median of 3,2,5,6,4,5,3,5 is 4.5 because 4.5 is the averageof the two middle numbers of the sorted data i.e.2,3,3,4,5,5,5,6.

Measure of the Central Tendency of a Data Set

Mode: Observation in the data set with the largest frequency.Note that we can have more than one mode for a data set.

Mode: Example

For example the mode of 0,5,1,1,3 is 1.

Effect of an Outlier

Calculate mean, median, and mode of 0,5,1,1,3,100.

mean=18.33, median=2, mode=1.

Effect of an Outlier

Outlier pulls mean towards it but may not affect median andmode.

Identifying Relation Between Mean and Median

from Histogram


from Histogram

Symmetric: mean median


from Histogram

Symmetric: mean median

Left skewed: Mean < Median < Mode (in general)

Right skewed: Mean > Median > Mode (in general)

Kurtosis

Leptokurtic

Platykurtic

Mesokurtic

Measure of the Spread of a Data Set

Range: max-min

Ex: 0,5,1,1,3; what is the range?

Range = 5 0 = 5.


Range: max-min

Ex: 0,5,1,1,3; what is the range?Range = 5 0 = 5.


Variance:n

i=1(xix)2n1

Standard deviation:n

i=1(xix)2n1

Variance and Standard Deviation: Example

What is the variance and the standard deviation of 3,3,3,3,3?

Ans. variance=0 standard deviation=0

What is the variance and the standard deviation of 1,2,3,4,5?Ans. variance=2.5 standard deviation=1.58


What is the variance and the standard deviation of 3,3,3,3,3?Ans. variance=0 standard deviation=0

What is the variance and the standard deviation of 1,2,3,4,5?

Ans. variance=2.5 standard deviation=1.58


What is the variance and the standard deviation of 3,3,3,3,3?Ans. variance=0 standard deviation=0

What is the variance and the standard deviation of 1,2,3,4,5?Ans. variance=2.5 standard deviation=1.58

Standard Deviation

Standard deviation is always greater than or equal to zero.

Does Standard Deviation Gets Affected by

Outliers?

What is the standard deviation for the data 3,3,3,3,100?

Ans. 43.38

Is Standard Deviation Always a Good Measure of

the Spread of a Data Set?

Not a good measure when data is skewed or has outliers.

Quartiles

First quartile: 25th percentile

Notation: Q1

Quartiles

Third quartile: 75th percentile

Notation: Q3

Exercise

Find the first and third quartile of 8,7,1,4,6,6,4,5,7,6,3,0.

Ans. The sorted data is 0,1,3,4,4,5,6,6,6,7,7,8. The median ofthe red half of the data is 3.5 (Q1) and the median of the bluehalf of the data is 6.5 (Q3).

Quartiles

Median is the second quartile (Q2).


Inter Quartile Range (IQR): Q3 Q1

*IQR is a robust measure of spread. IQR does not get affectedmuch by skewness or outliers.

Exercise

Find IQR of 8,7,1,4,6,6,4,5,7,6,3,0.

Q3-Q1=6.5-3.5=3.

Five Number Summary

Minimum

First quartile

Median

Third quartile

Maximum

Boxplot

*We will create a box plot for the Cafe data set.

Interpreting a Box Plot

Shape:

Outliers: Any observation not in the range[Q1 1.5 IQR,Q3 + 1.5 IQR] is considered an outlier(Informal Rule).

Why Do We Need Box Plot?

To compare two or more data sets.

Visualization of summary statistics.

Categorical Data Visualization

*Bar Chart

*Pie Chart

Show billionaires data.

Part 2: Introduction to Probability

Describing Shape of a Bar Graph

Proportion of observations in a particular category.

Describing Shape of a Histogram

Proportion of observations in a particular class interval.

Probability

Proportion sample

Probability population

Example

Workforce distribution in the United States.

Industry ProbabilityAgriculture 0.130Construction 0.147Finance, Insurance, Real Estate 0.059Manufacturing 0.042Mining 0.002Services 0.419Trade 0.159Transportation, Public Utilities 0.042

Sample Space

Def: Set of all possible outcomes.

E.g.: ={Agriculture, Construction, . . . , Services, Trade,Transportation and Public Utilities}

Simple Events

Simple event: An event in the finest partition of the samplespace.

Example: 1=Agriculture, 2=Construction.

Event

Def: Any subset of the sample space

E.g.: {Agriculture, Construction}

Exercise

A bowl contains three red and two yellow balls. Two balls arerandomly selected and their colors recorded. Use a treediagram to list the 20 simple events in the experiment, keepingin mind the order in which the balls are drawn.

Other Approaches for Calculating Probabilities

Classical Approach: Assuming all outcomes to be equallylikely, the probability of an event is the number of favourableoutcomes divided by the total number of outcomes.E.g. Rolling a dice

Subjective Approach: Assigning probability to an event basedon ones experience.

Example

Workforce distribution in the United States.

Industry ProbabilityAgriculture 0.130Construction 0.147Finance, Insurance, Real Estate 0.059Manufacturing 0.042Mining 0.002Services 0.419Trade 0.159Transportation, Public Utilities 0.042

Probability

P(Agriculture)

= 0.13

P(Either Agriculture or Construction or both) P(Agriculture Construction) = 0.13+0.147=0.277.

P(Agriculture and Construction) P(Agriculture Construction) =0.

P(Not in Agriculture) P(Agriculturec) = 1-0.13=0.87.

Probability

P(Agriculture) = 0.13

P(Either Agriculture or Construction or both) P(Agriculture Construction)

= 0.13+0.147=0.277.



Probability



P(Agriculture and Construction) P(Agriculture Construction)

=0.


Probability




P(Not in Agriculture) P(Agriculturec)

= 1-0.13=0.87.

Probability





Compound Events

If A and B are two events then

Union event is A B

Intersection event is A B

Complement event is Ac

Venn Diagram Representation

8

A B

S

Disjoint events A and B A B

A

S

B

U

A U B

A

S

B

C

BS

Mutually exclusive and exhaustiveevents: A, B, C, and D

A

D

Probability Rules

1 P(A B) = P(A) + P(B) P(A B)2 P(Ac) = 1 P(A)

Mutually Exclusive

Def: Two events are mutually exclusive if they do not haveany common outcome.

E.g.: Agriculture and Construction are mutually exclusiveevents.

Mutually Exclusive

A and B are mutually exclusive if P(A B) = 0.

This implies that for mutually exclusive events A and B,P(A B) = P(A)+P(B).

Pizza Venn Diagram

What is the sample space?

Sample space={Tomato only, Fish Only, Mushroom-Tomato,Mushroom-Tomato-Fish, Mushroom-Fish, No toppings}.

Probability of the events in the sample space

P(Tomato only)

=2/8; P(Fish only)=1/8.

P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.

P(Mushroom-Fish) =1/8; P(No toppings)=1/8.


P(Tomato only) =2/8; P(Fish only)

=1/8.




P(Tomato only) =2/8; P(Fish only)=1/8.

P(Mushroom-Tomato)

=2/8=1/4;P(Mushroom-Tomato-Fish)=1/8.




P(Mushroom-Tomato) =2/8=1/4;P(Mushroom-Tomato-Fish)

=1/8.





P(Mushroom-Fish)

=1/8; P(No toppings)=1/8.




P(Mushroom-Fish) =1/8; P(No toppings)

=1/8.

Union Rule

What is the probability that your slice will have tomato ormushroom?

Ans. 6/8=3/4

Intersection Rule

What is the probability that your slice will have tomato andmushroom?

Ans. 3/8

Complement Rule

What is the probability that your slice will not have tomato?

Ans. 3/8

Conditional Probability

You have pulled out a slice of pizza that has tomato on it.What is the probability that your slice will have mushrooms?

Ans. 3/5.

Conditional Probability

Def: Probability of event A in event B. That is, probabilitythat even A occurs given than B occurs.

Notation: A|B

Multiplication rule

P(A B) = P(A)P(B |A)P(A B) = P(B)P(A|B)

Statistical Independence

Two events are said to be independent if the occurrence ofone has no effect on the chance of occurrence of the other.


Two events A and B are considered independent whenP(A|B)=P(A).

Exercise 1

Is gender related to whether someone voted in the last mayoralelection? Answer the question using the joint probabilitiesgiven in the table below.

GenderVoted in the last mayoral election Female MaleYes 0.25 0.18No 0.33 0.24


If two events A and B are independent then

1 P(A B) = P(A)P(B)

Law of Total Probability

Given a set of events S1, S2, . . . , Sk that are mutually exclusiveand exhaustive, and an event A, the probability of the event Acan be expressed as

P(A) = P(S1).P(A|S1) + P(S2).P(A|S2)+P(S3).P(A|S3) + . . . + P(Sk).P(A|Sk)

Exercise 2

A business group owns three five-star hotels (say, A, B, and C)in India. By studying the past behavior of the revenueobtained from the three hotels month by month, it has beenobserved that the probability of increase in revenue of either Bor C or both of them is 0.5. If As revenue increases in a givenmonth, the probability of increase in Bs revenue is 0.7, theprobability of increase in Cs revenue is 0.6, and the probabilityof increase in both B and Cs revenue is 0.5. However if Asrevenue does not increase in a given month, the probability ofincrease in Bs revenue is 0.2, the probability of increase in Csrevenue is 0.3, and the probability of increase in both B andCs revenue is 0.1. What is the probability that the revenue ofall the three hotels, A, B, and C, increase in a given month?

Exercise 3

You are a physician. You think it is quite likely that one of your patients has strep

throat, but you are not sure. You take some swabs from the throat and send them to

a lab for testing. The test is (like nearly all lab tests) not perfect. If the patient has

strep throat, then 70% of the time the lab says YES but 30% of the time it says NO.

If the patient does not have strep throat, then 90% of the time the lab says NO but

10% of the time it says YES. You send five succesive swabs to the lab, from the same

patient. You get back these results, in order; YNYNY. What do you conclude?

These results are worthless.

It is likely that the patient does not have the strep throat.

It is slightly more likely than not, that patient does have the strep throat.

It is very much more likely than not, that patient does have the strep throat.

Bayes Rule

Let S1, S2, . . . , Sk represents k mutually exclusive andexhaustive sub-populations with prior probabilitiesP(S1),P(S2), . . . ,P(S2). If an event A occurs, the posteriorprobability of Si given A is the conditional probability

P(Si |A) = P(Si).P(A|Si)kj=1 P(Sj).P(A|Sj)

Exercise

Strep Throat Exercise

Bibliography

An Introduction to Probability and Inductive Logic, by IanHacking

Introduction to Probability and Statistics, by WilliamMendenhall, Robert J. Beaver, and Barbara M. Beaver

Practice of Business Statistics, by David S. Moore, GeorgeP. McCabe, William M. Duckworth, and Stanley L. Sclove

Bradley A. Warner, David Pendergrift, and TimothyWebb,That was Venn, This is now, Journal ofStatistical Education, Volume 6, Number 1, 1998

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