statistical characterization

8/9/2019 Statistical Characterization

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STATIST

ICAL

CHAR

ACTE

RIZATION


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E*ample o$ Arithmetic Mean :

The salaries o$ +"e teachers are as $ollo,s& -in! the mean salary

../001 .23001 ./0001 .3/001 .3400&

Soltion:

.5630 Rpees


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7ariance The 7ariance is !e+ne! as a"era%e o$ the s8are! !i9erences $rom

the Mean&A measrement o$ the sprea! #et,een nm#ers in a !ataset& The "ariance measres ho, $ar each nm#er in the set is $romthe mean& 7ariance is !enoate! #y 2.

Here

2= variance

(X - )2= The sum of (X - )2 for all datapoints

X = individual data points 0r random variale

= mean of the population

! = numer of data points


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To calclate the "ariance $ollo, these steps:

Calclate the Mean(the simple a"era%e o$ the nm#ers)

Then $or each nm#er: s#tract the Mean an! s8are the reslt

Then ,or ot the a"era%e o$ those s8are! !i9erences& ('hy S8are;)

E*ample


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-irst step is to +n! ot the mean

so the mean (a"era%e) hei%ht is 5?3 mm& Let@s plot this on the chart:

No, ,e calclate each !o%@s !i9erence $rom the Mean:


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So the 7ariance Is 21,704


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So the 7ariance is 223&5


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Co"ariance :Co"ariance is one o$ the statistical measrement to no, the relationship o$ the

"ariances #et,een the t,o "aria#les& It helps s to no, ,hether the t,o "aria#les"ary to%ether or chan%e to%ether&

Here the si%n o$ co"ariance tells s the natre o$ the relationship o$ the "ariances& I$the co"arience is positi"e1 then the t,o "aria#les x an! ymo"e in the same!irection& I$ its ne%ati"e then the "aria#les mo"e in opposite !irections&

In the same ,ay1 sie o$ the co"ariance helps s to no, the stren%th o$ therelationship& I$ the co"ariance is lar%e1 then there is a stron% relationship1 i$ its small1then there is a ,ea or no relationship ,ith the t,o "aria#les&

The Co"ariance is !enote! as Cov(X,Y) an! is %i"en as1


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Correalation

The ,or! Correlation is ma!e o$ Co-(meanin% Bto%etherB)1 an! Relation.

Correlation is a statistical techni8e that can sho, ,hether an! ho, stron%ly pairs o$

"aria#les are relate!& 'hen t,o sets o$ !ata are stron%ly line! to%ether ,e saythey ha"e a Hi%h Correlation

Correlation is Positive,hen the "ales increaseto%ether&

Correlation is Neative,hen one "ale !ecreasesas the other increases&

Lie this


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Correlation coecientThe most $amiliar measre o$ !epen!ence #et,een t,o 8antities is the Dearson@s

correlation coecientB1 commonly calle! correlation coecient&

The main reslt o$ a correlation is calle! the correlation coe#cient(or BrB)& It ran%es$rom F.&0 to G.&0& The closer r is to G. or F.1 the more closely the t,o "aria#les arerelate!&

I$ ,e ha"e a series o$ nmeasrements o$Xan! ",ritten as#ian!$i,here i .1

21 &&&1 n1 then the sample correlation coe%cientcan #e se! to estimate thepoplation earson correlation r#et,eenXan! "& The sample correlation coecient

is ,ritten as

or

E l I C S l


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E*ample: Ice Cream Sales

The local ice cream shop eeps trac o$ ho, mch ice cream they sell "erss thetemperatre on that !ay1 here are their +%res $or the last .2 !ays&

'hat ,ill #e the correlation;

Ice Cream Sales vs Temperature

Temperatre C Ice Cream Sales

.3&2 2./

.6&3 52/

..&? .4/

./&2 552

.4&/ 306

22&. /22

.?&3 3.2

2/&. 6.3

25&3 /33

.4&. 32.

22&6 33/

.>&2 304


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St t l th l ti i t


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Steps to sol"e the correlation coecient

Let s call the t,o sets o$ !ata B*B an! ByB (in or case Temperatre is xan! Ice CreamSales is y):

Step .: -in! the mean o$ x1 an! the mean o$ y

Step 2: S#tract the mean o$ * $rom e"ery * "ale (call them BaB)1 !o the same $or y(call them B$B)

Step 5: Calclate: a % $1 a2an! $2$or e"ery "ale

Step 3: Sm p a % $1 sm p a2an! sm p $2

Step /: Ji"i!e the sm o$ a K # #y the s8are root o$ (sm o$ a2) K (sm o$ #2)

A$ter this ,e ,ill pt all "ales in the $ormla to +not the correlation


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statistical characterization

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