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Business Statistics

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Why statistics?

Decision making is often based on

analysis of data.

Statistics helps you to make sense of thedata by using tools that summarize,

present and analyze the data.

Decision maker can also ascertain the

confidence in the decisions.

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Eamples

!o" many ne"spapers should the #endor stock

to maimize re#enue?

$ Depends on the probability distribution of demand and

epected profit %re t"o or more market segments significantly

different?

$ !ypothesis testing

What proportion of people are happy "ith the

Sith&pay commission report?

$ 'arameter estimation

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Sample #s. 'opulation

'opulation is the entire group(collection ofindi#iduals(ob)ects(things that "e "antinformation about.

Sample is part of the population that "e actuallyeamine to gather information.

Eample$ We "ish to find the a#erage di#idend percentage of

all companies traded at *SE. %ll stocks traded at *SE comprises population

+- of the stocks selected for gathering information is thesample

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Inferential Statistics Predict and forecast

values of populationparameters

Test hypotheses about

values of population

parameters

Make decisions

Descriptive Statistics Collect

Organize

Summarize

Display

nalyze

Subdivision within Statistics

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Descripti#e statistics

& data and freuency distribution /he follo"ing are the departure delay in minutes of 01 flights selected

at random from a particular airport.

+ +1 02

+3 4 0

+3

1 02

52 34 67

0 07 22

26 2

02 2 17

2 +2 16

30 +1 12

04 0 12

2 01 04

23 00 13

26 06 11

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8reuency Distribution

/able "ith t"o columns listing9

Each and e#ery group or class or inter#al of #alues

%ssociated frequency of each group

*umber of obser#ations assigned to each group Sum of freuencies is number of obser#ations

:lassmidpoint is the middle #alue of a group or class or

inter#al

Relative frequencyis the percentage(proportion of totalobser#ations in each class

Sum of relati#e freuencies ; +

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8reuency distribution

Delay inminutes

8reuency

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8reuency distribution& histogram

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/"o #ariable freuency distribution

&cross tabulation

% )oint freuency distribution of t"o #ariables =e.g. o"nership of airline, delay

in minutes>

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Descripti#e statistics & measures

easures of @ocation

easures of Aariability

Ske"ness and urtosis%ssociation bet"een t"o #ariables

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easures of @ocation

%rithmetic ean

edian

ode 'ercentiles

Cuartiles

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%rithmetic mean

/he mean of a data set is the a#erage

of all the data #alues.

x xn

i=x xn

i=

=xN

i=xN

i

Sample mean

'opulation mean

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ean $ eample

%#erage delay in flight departure

xx; +320(01 ; 31.134+ minutes

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edian

t is the middle item in a data set that isarranged in ascending(descending order

f there are n obser#ations then the

edian ; =n+>(1 th obser#ation.

computation rule

if n is odd then =n+>(1 is an integer if n is e#en then use a#erage of n(1 and n(1 + th

obser#ation

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Eample

Sorted 01

obser#ations

median is a#erage of

1+stand 11ndobser#ation

; =3034>(1

; 36

11 02 13 06

12 07

12 04

0 16 04

2 17 2

4 34 2

+ 38 2

+1 0 23

+1 0 22

+3 01 26

+3 00 26

+2 02 67

1 02 52

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ode

ode is the highest occurring obser#ation

$ mode in the eample is

/he greatest freuency can occur at t"oor more different #alues.

f the data ha#e eactly t"o modes, the

data are bimodal.

f the data ha#e more than t"o modes, the

data are multimodal.

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Fi#en any set of ordered numerical

obser#ations /he Pth percentilein the orderedset is that

#alue belo" "hich lie P- =Ppercent> of the

obser#ations in the set.

/he positionof the Pthpercentile is gi#en by (n+

1)P1!!, "here nis the number of obser#ations inthe set.

'ercentiles and Cuartiles

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Eample

:alculate 02thpercentile of the airline

delay data

the position of 02thpercentile is

02G=01+>(+ ; +5.32th

#alue of 02thpercentile

; +5th

obser#ation .32 of =1 $ +5>thobser#ation

; 16.32 =16 .32=17&16>>

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Cuartiles

Cuartiles are special names to percentiles

C+ ; 12thpercentile

C1 ; 2th

percentile ; median C3 ; 72thpercentile

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easures of Aariability

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nteruartile range

/he interuartile range of a data set is the

difference bet"een the third uartileand the first

uartile.

t is the range for the middle 2- of the data. t o#ercomes the sensiti#ity to etreme data

#alues.

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Aariance

/he #ariance is a measure of #ariability

that utilizes all the data.

t is based on the difference bet"een the

#alue of each obser#ation =xi> and the

mean =xfor a sample, for a population>.

2

2

= ( )xNi 2

2

= ( )xNi s xi x

n2

2

1=

( )s xi x

n2

2

1=

( )H & 'opulation #ariance

Sample #ariance & I

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Standard de#iation

/he standard de#iation of a data set is thepositi#e suare root of the #ariance.

t is measured in the same units as the

data, making it more easily comparable,than the #ariance, to the mean.

f the data set is a sample, the standard

de#iation is denoted s. f the data set is a population, the standard

de#iation is denoted =sigma>.

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:oefficient of Aariation

/he coefficient of #ariation indicates ho" large the

standard de#iation is in relation to the mean. f the data set is a sample, the coefficient of #ariation

is computed as follo"s9

f the data set is a population, the coefficient of

#ariation is computed as follo"s9

s

x ( )100

s

x ( )100

( )100

( )100

s

x ( )100

s

x ( )100

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Eample

Aariance

; 062.45 minutes suare

Standard De#iation

; 1+.242 minutes

:oefficient of Aariation ;

; 1+.240(31.134+ =+> ; 66.52-

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S"ewness

$ Ske"ness characterizes the degree of

asymmetry of a distribution around its

mean 'ositi#ely ske"ed

Symmetric or unske"ed

*egati#ely ske"ed

Ske"ness

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!egatively ske"ed

Ske"ness

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Ske"ness

Symmetric

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Ske"ness

Positively Ske"ed

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Ske"ness & measure

3

3

1

)(

N

X=

Ske"ness of a distribution is measured by

8or a gi#en data set you may use

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urtosis

urtosis characterizes the relati#e

peakedness or flatness of a symmetric

distribution compared to the normal

distribution

'latykurtic=relati#ely flat>

esokurtic=normal>

@eptokurtic=relati#ely peaked>

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urtosis

Platykurtic- flat distribution

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urtosis

Mesokurtic - not too flat and not too peaked

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urtosis

#eptokurtic- peaked distribution

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urtosis & measure

urtosis for a distribution is measured by

4

4

2

)(

N

X=

31

=

"here

8or a gi#en data set you may use

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%ssociation bet"een t"o #ariables

#elay $assen%ers #elay $assen%ers #elay $assen%ers

23 62 26 2+ 2 64

0 6+ 01 2 71

06 23 12 27 34 70

62 +3 27 22 64

11 02 0 20 02 73

2 24 4 20 +2 63

00 64 17 62 04 64

+1 62 67 27 22

+1 26 04 61 + 02

12 2 0 2 2 7+

+3 7 02 6+ 26 60

2 73 25 16 6

02 63 30 63 07 6+

13 26 52 05 1 04

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%ssociation bet"een t"o #ariables

Scatter plot

:o#ariance

:orrelation :oefficient

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Scatter 'lot

Scatter $lotsare used to identify any

underlying relationships among pairs of

data sets.

/he plot consists of a scatter of points,

each point representing an obser#ation.

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Scatter 'lot

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:o#ariance

/he co#ariance is a measure of the linear

association bet"een t"o #ariables.

'ositi#e #alues indicate a positi#e

relationship.

*egati#e #alues indicate a negati#e

relationship

: i

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f the data sets are samples, the co#ariance

is denoted by

f the data sets are populations, theco#ariance is denoted by

:o#ariance

s x x y y

nxy

i i=

( )( )

1s

x x y y

nxy

i i=

( )( )

1

xy i x i yx yN

=

( )( )

xy i x i yx y

N=

( )( )

; 1.01 in the

%irline

eample

: l ti : ffi i t

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:orrelation :oefficient

/he coefficient can take on #alues bet"een &+ and +.

Aalues near &+ indicate a strong negati#e linear relationship. Aalues near + indicate a strong positi#e linear relationship.

f the data sets are samples, the coefficient is

f the data sets are populations, the coefficient is

xyxy

x y=

xyxy

x y=

rs

s sxy xy

x y=r

s

s sxy xy

x y= ; .+1+ in %irlineeample