measures of dispersion and some basic probability

8/10/2019 Measures of Dispersion and Some Basic Probability

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2003 Prentice-Hall, Inc. Chap 3-1

Business Statistics

Dispersion measures and Some BasicProbability Concepts


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Chapter Topics

Measure of Variation

Range, Variance and Standard Deviation, Coefficient of

Variation

Shape

Symmetric, Skewed

Basic Probability Concepts

Sample spaces and events, simple probability, jointprobability

Conditional Probability

Statistical independence, marginal probability


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Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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Measures of Variation

Variation

Variance Standard Deviation Coefficient

of VariationPopulation

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range

Interquartile Range


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Range

Measure of Variation

Difference between the Largest and the

Smallest Observations:

Ignores How Data are Distributed

Largest SmallestRange X X

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5


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2

2 1

N

i

i

X

N

Important Measure of Variation

Shows Variation About the Mean

Sample Variance:

Population Variance:

2

2 1

1

n

i

i

X X

Sn

Variance


7/32 2003 Prentice-Hall, Inc. Chap 3-7

Standard Deviation

Most Important Measure of Variation

Shows Variation about the Mean

Has the Same Units as the Original Data

Sample Standard Deviation:

Population Standard Deviation:

2

1

1

n

i

i

X X

Sn

2

1

N

i

i

X

N



Comparing Standard Deviations

Mean = 15.5

s = 3.33811 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C



Coefficient of Variation

Measure of Relative Variation

Always in Percentage (%)

Shows Variation Relative to Mean

Used to Compare Two or More Sets of Data

Measured in Different Units

100%

SCV

X



Shape of a Distribution

Describe How Data are Distributed

Measures of Shape

Symmetric or skewed

Mean =Median =ModeMean



Some Basic Probability Concepts



Sample Spaces

Collection of All Possible Outcomes

e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:



Events

Simple Event

Outcome from a sample space with 1 characteristic

e.g. A Red Card from a deck of cards Joint Event

Involves 2 outcomes simultaneously

e.g. An Ace which is also a Red Card from a deckof cards



Visualizing Events

Contingency Tables

Tree Diagrams

Red 2 24 26Black 2 24 26

Total 4 48 52

Ace Not Ace Total

FullDeck

of Cards

Red

Cards

Black

Cards

Not an Ace

Ace

Ace

Not an Ace



Simple Events

The Event of a Happy Face

There are 5happy faces in this collection of 18objects



The Event of a Happy Face AND Yellow

Joint Events

1Happy Face which is Yellow


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Special Events

Impossible Event

Impossible event

e.g. Club & Diamond on 1 carddraw

Complement of Event

For event A, all events not in A

Denoted as A

e.g. A: Queen of DiamondA: All cards in a deck that are not Queen of

Diamond

Impossible Event


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Special Events

Mutually Exclusive Events

Two events cannot occur together

e.g. A: Queen of Diamond; B: Queen of Club

Events A and B are mutually exclusive Collectively Exhaustive Events

One of the events must occur

The set of events covers the whole sample space

e.g. A: All the Aces; B: All the Black Cards; C: All theDiamonds; D: All the Hearts

Events A, B, C and D are collectively exhaustive

Events B, C and D are also collectively exhaustive

(continued)


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Contingency Table

A Deck of 52 Cards

AceNot an

Ace Total

Red

Black

Total

2 24

2 24

26

26

4 48 52

Sample Space

Red Ace


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FullDeck

of Cards

Tree Diagram

Event Possibilities

RedCards

Black

Cards

Ace

Not an Ace

Ace

Not an Ace


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Probability

Probability is the Numerical

Measure of the Likelihood

that an Event Will Occur

Value is Between 0 and 1

Sum of the Probabilities of

all Mutually Exclusive and

Collective Exhaustive Events

is 1

Certain

Impossible

.5

1

0


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(There are 2 ways to get one 6 and the other 4)

e.g. P( )= 2/36

Computing Probabilities

The Probability of an Event E:

Each of the Outcomes in the Sample Space is

Equally Likely to Occur

number of event outcomes( )

total number of possible outcomes in the sample space

P E

X

T


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Computing Joint Probability

The Probability of a Joint Event, A and B:

( and )

number of outcomes from both A and Btotal number of possible outcomes in sample space

P A B

E.g. (Red Card and Ace)2 Red Aces 1

52 Total Number of Cards 26

P


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P(A1and B2) P(A1)TotalEvent

Joint Probability UsingContingency Table

P(A2 and B1)

P(A1and B1)

Event

Total 1

Joint Probability Marginal (Simple) Probability

A1

A2

B1 B2

P(B1) P(B2)

P(A2and B2) P(A2)


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Computing CompoundProbability

Probability of a Compound Event, A or B:

( or )

number of outcomes from either A or B or both

total number of outcomes in sample space

P A B

E.g. (Red Card or Ace)

4 Aces + 26 Red Cards - 2 Red Aces52 total number of cards

28 7

52 13

P


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P(A1)

P(B2)

P(A1and B1)

Compound Probability(Addition Rule)

P(A1or B1) = P(A1)+ P(B1)- P(A1and B1)

P(A1and B2)TotalEvent

P(A2 and B1)

Event

Total 1

A1

A2

B1 B2

P(B1)

P(A2and B2) P(A2)

For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

d l


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Computing ConditionalProbability

The Probability of Event A given that Event BHas Occurred:

( and )( | )( )

P A BP A B

P B

E.g.

(Red Card given that it is an Ace)

2 Red Aces 1

4 Aces 2

P

C di i l b bili i


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Conditional Probability UsingContingency Table

Black

ColorType Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Revised Sample Space

(Ace and Red) 2 / 52 2(Ace | Red)

(Red) 26 / 52 26

PP

P

C di i l P b bili d


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Conditional Probability andStatistical Independence

Conditional Probability:

Multiplication Rule:

( and )

( | ) ( )

P A BP A B

P B

( and ) ( | ) ( )

( | ) ( )

P A B P A B P B

P B A P A

C diti l P b bilit d


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Conditional Probability andStatistical Independence

Events A and B are Independent if

Events A and B are Independent when theProbability of One Event, A, is Not Affected by

Another Event, B

(continued)

( | ) ( )

or ( | ) ( )

or ( and ) ( ) ( )

P A B P A

P B A P B

P A B P A P B


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Chapter Summary

Described Measures of Variation

Range, Variance and Standard Deviation,

Coefficient of variation Illustrated Shape of Distribution

Symmetric, Skewed


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Chapter Summary

Discussed Basic Probability Concepts

Sample spaces and events, simple probability, and

joint probability

Defined Conditional Probability

Statistical independence, marginal probability

measures of dispersion and some basic probability

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