statistic copy 1

8/18/2019 Statistic Copy 1

1/69

Note: Most of the Slides were taken from

Elementary Statistics: A Handbook of Slide

Presentation prepared by Z.V.. Albacea! ".E.

#eano! #.V. "ollado! $.N. "omia and N.A.

%andan& in '(() for the *nstit+te of Statistics!

"AS! ,P $os -anos

Training on Teaching

Basic Statistics for

Tertiary Level Teachers

Summer 2008

INTRODUCTION TO

STATISTICS AND

STATISTICALINFERENCE


2/69

Session 1.2

TEACHING BASIC STATISTICS ….


3/69

Session 1.3



4/69

Session 1.4



5/69

Session 1.5



6/69

Session 1.6



7/69

Session 1.7



8/69

Session 1.8



9/69

Session 1.9


Areas of Statistics

Descriptive statisticsmethods concerned w/

collecting descri!ing andanal"#ing a set of data

witho$t drawing

concl$sions %or inferences&

a!o$t a large gro$'

Inferential statisticsmethods concerned

with the anal"sis of as$!set of data leading

to 'redictions or

inferences a!o$t the

entire set of data


10/69

Session 1.10


Example of Descriptive Statistics

(resent the (hili''ine 'o'$lation !" constr$cting a

gra'h indicating the total n$m!er of Fili'inos co$nted

d$ring the last cens$s !" age gro$' and se)


11/69

Session 1.11


Example of Inferential Statistics

A new milk form+lation desi&ned to improe the psychomotordeelopment of infants was tested on randomly selected infants.

-ased on the res+lts! it was concl+ded that the new milk form+lation is

effectie in improin& the psychomotor deelopment of infants*


12/69

Session 1.12


Inferential Statistics

Larger Set(N units/observations) Smaller Set

(n

units/observations)

Inferences and

Generalizations


13/69

Session 1.13


+e" Definitions

A +nierse is the collection of things or

o!ser,ational $nits $nder consideration*

A ,aria!le is a characteristic o!ser,edor meas$red on e,er" $nit of the

$ni,erse*

A 'o'$lation is the set of all 'ossi!le,al$es of the ,aria!le*


14/69

Session 1.14


+e" Definitions

Parameters are n$merical meas$res

that descri!e the 'o'$lation or $ni,erse

of interest* Us$all" donated !" -ree.letters µ %m$& σ %sigma& ρ %rho& λ

%lam!da& τ %ta$& θ %theta& α %al'ha& and

β %!eta&*

Statistics are n$merical meas$res of a

sam'le


15/69

Session 1.15


T"'es of 0aria!les

1$alitati,e ,aria!le non2n$merical ,al$es

1$antitati,e ,aria!le n$merical ,al$es

a* Discrete co$nta!le

!* Contin$o$s meas$ra!le

c* Constant


16/69

Session 1.16


Le,els of 3eas$rement4* Nominal

N$m!ers or s"m!ols $sed to classif"

5* Ordinal scale Acco$nts for order no indication of distance

!etween 'ositions Used in ran.ing no meaningf$l n$merical

statements can !e made a!o$t difference !etweencategories*

6* Inter,al scale E7$al inter,als no a!sol$te #ero

8* Ratio scale 9as a!sol$te #ero


17/69

N!IN"L #en$er% &olitical &arty%'eligion% "utomo(ile)nership%

ORDINAL Teachers (erformance 3o,ieClassification Fac$lt" Ran.9otel Ratings St$dent Class

Designation

INTER0AL Tem'erat$re

RATIO :eight Age Salar"

Session 1.17



18/69

Session 1.18


!etho$s of *ollecting Data

(+ective !etho$

Su(+ective !etho$

,se of Existing 'ecor$s


19/69

Session 1.19


!etho$s of &resenting Data

Textual

Ta(ular

#raphical


20/69

Session 1.20


Mean

Median

Mode

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range

Location

Maximum

Minimum

Central

Tendency

PercentileQuartile

Decile

Interquartile

Range

Se!ness

"urtosis


21/69

Session 1.21


!easures of Location

A Measure of Location s$mmari#es adata set !" gi,ing a ;t"'ical ,al$e< within

the range of the data ,al$es that descri!esits location relati,e to entire data set*

Some Common 3eas$res=

3inim$m 3a)im$m

Central Tendenc"

(ercentiles Deciles 1$artiles


22/69

Session 1.22


!aximum an$ !inimum

Minimum is the smallest ,al$ein the data set denoted as M*N *

Maximum is the largest ,al$e inthe data set denoted as MA/ *


23/69

Session 1.23


3eas$re of Central Tendenc"

A single ,al$e that is $sed to identif"

the ;center< of the data

it is tho$ght of as a t"'ical ,al$e ofthe distri!$tion

'recise "et sim'le

most re'resentati,e ,al$e of thedata


24/69

Session 1.24


3ean

3ost common meas$re of the center

Also .nown as arithmetic a,erage

Sam#le Mean

Po#ulation Mean


25/69

Session 1.25


(ro'erties of the 3ean

ma" not !e an act$alo!ser,ation in the data set

can !e a''lied in at leastinter,al le,el

eas" to com'$te

e,er" o!ser,ation contri!$tes tothe ,al$e of the mean


26/69

Session 1.26


(ro'erties of the 3ean

s$!gro$' means can !e com!ined

to come $' with a gro$' mean

easil" affected !" e)treme ,al$es

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean $ %Mean $ &


27/69

Session 1.27


3edian

Di,ides the o!ser,ations into two e7$al

'arts

If the n$m!er of o!ser,ations is odd themedian is the middle n$m!er*

If the n$m!er of o!ser,ations is e,en the

median is the a,erage of the 5 middle

n$m!ers*

Sam'le median denoted as

while 'o'$lation median is denoted as

x~

µ ~


28/69

Session 1.28


(ro'erties of a 3edian

ma" not !e an act$al o!ser,ation inthe data set

can !e a''lied in at least ordinal le,el a 'ositional meas$re not affected !"e)treme ,al$es

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median $ %


29/69

Session 1.29


3ode

occ$rs most fre7$entl"

nominal a,erage

ma" or ma" not e)ist

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode $ '

0 1 2 3 4 5 6

(o Mode


30/69

Session 1.30


(ro'erties of a 3ode

can !e $sed for 7$alitati,e as

well as 7$antitati,e data ma" not !e $ni7$e not affected !" e)treme ,al$es can !e com'$ted for $ngro$'ed

and gro$'ed data


31/69

Session 1.31


3ean 3edian > 3ode

Use the mean when=

sam'ling sta!ilit" is desired other meas$res are to !e

com'$ted


32/69

Session 1.32


3ean 3edian > 3ode

Use the median when=

the e)act mid'oint of thedistri!$tion is desired

there are e)treme

o!ser,ations


33/69

Session 1.33


3ean 3edian > 3ode

Use the mode when=

when the ?t"'ical? ,al$e isdesired

when the dataset is meas$red

on a nominal scale


34/69

Session 1.34


&ercentiles

N$merical meas$res that gi,e therelati,e 'osition of a data ,al$erelati,e to the entire data set*

Di,ide an arra" %raw data arran&edin increasin& or decreasin& order ofma&nit+de& into 4@@ e7$al 'arts*

The 0 th 'ercentile denoted as P j , isthe data ,al$e in the the data setthat se'arates the !ottom 0 of thedata from the to' %4@@2 0 &*


35/69

Session 1.35


E-"!&LE

S$''ose LB was told that relati,eto the other scores on a certain

test his score was the th

'ercentile*

%his means that 1)2 of those

who took the test had scores lessthan or e3+al to $4s score! while)2 had scores hi&her than $4s.


36/69

Session 1.36


Deciles

Di,ide an arra" into ten e7$al'arts each 'art ha,ing ten

'ercent of the distri!$tion of thedata ,al$es denoted !" 5 0 *

The 4st decile is the 4@th 'ercentile the 5nd decile is the5@th 'ercentile**


37/69

Session 1.37


.uartiles

Di,ide an arra" into fo$r e7$al'arts each 'art ha,ing 5 of

the distri!$tion of the data,al$es denoted !" 6 0 *

The 4st 7$artile is the 5th 'ercentile the 5nd 7$artile is

the @th

'ercentile also themedian and the 6rd 7$artile isthe th 'ercentile*

TEACHING BASIC STATISTICS


38/69

Session 1.38


3eas$res of 0ariation

A meas$re of ,ariation is a

single ,al$e that is $sed to

descri!e the s'read of the

distri!$tion A meas$re of central tendenc"

alone does not $ni7$el"

descri!e a distri!$tion



39/69

Session 1.39


!ean / 11 s / 3338 2 3 4 1 5 6 8 7 20 2

2 3 4 1 5 6 8 7 20 2

Data B

Data "

!ean / 11

s / 7218

2 3 4 1 5 6 8 7 20 2

!ean / 11

s / 416

Data *

" loo at $ispersion9



40/69

Session 1.40


T)o Types of !easures of

Dispersion Absol+te Meas+res of 5ispersion:

Range

Inter27$artile Range 0ariance

Standard De,iation

#elatie Meas+re of 5ispersion: Coefficient of 0ariation



41/69

Session 1.41


'ange :';

The difference !etween the ma)im$m andminim$m ,al$e in a data set i*e*

R G 3AH 3INE7ample: ($lse rates of 4 male residents of a

certain ,illage

8 J J K@ K5 K KK 4

8 J J@ J5 J

# G J 2 8 G 64



42/69

Session 1.42


Some &roperties of the 'ange

The larger the ,al$e of therange the more dis'ersed

the o!ser,ations are* It is 7$ic. and eas" to

$nderstand*

A ro$gh meas$re ofdis'ersion*



43/69

Session 1.43


Inter


44/69

Session 1.44


Some &roperties of I.'

Red$ces the infl$ence of

e)treme ,al$es*

Not as eas" to calc$late

as the Range*



45/69

Session 1.45


0ariance

im'ortant meas$re of ,ariation

shows ,ariation a!o$t the mean

(o'$lation ,ariance

Sam'le ,ariance

N

X N

i

i∑=

−

= 1

2

2

)( µ

σ

1

)(1

2

2

−

−

=∑=

n

x x

s

n

i

i



46/69

Session 1.46


Standard De,iation %SD&

most im'ortant meas$re of ,ariation

s7$are root of 0ariance

has the same $nits as the original data

(o'$lation SD

Sam'le SD

N

X N

i

i∑=

−

= 1

2)( µ

σ

1

)(1

2

−

−

=

∑=

n

x x

s

n

i

i



47/69

Session 1.47


Data: 10 12 14 15 17 18 18 24

n = 8 Mean=16 Com'$tation of Standard De,iation



48/69

Session 1.48

Remar.s on Standard De,iation

If there is a large amo$nt of ,ariation

then on a,erage the data ,al$es will !e

far from the mean* 9ence the SD will !elarge*

If there is onl" a small amo$nt of

,ariation then on a,erage the data

,al$es will !e close to the mean* 9encethe SD will !e small*



49/69

Session 1.49

Mean $ )%*% s $ +*++,)) )- )+ ). )% )& )/ ), )' -0 -)

)) )- )+ ). )% )& )/ ), )' -0 -)

Data 1

Data 2

Mean $ )%*%

s $ *'-%,

)) )- )+ ). )% )& )/ ), )' -0 -)

Mean $ )%*%

s $ .*%/

Data C

Com'aring Standard De,iation



50/69

Session 1.50

Example: Team A - Heights of five maratho !"a#ers i i$hes

65 % 65 % 65 % 65 % 65 %

Mean $ &% S & 0




51/69

Session 1.51

Example: Team ' - Heights of five maratho !"a#ers i i$hes

62 % 67 % 66 % 70 % 60 %

Mean $ &%3 s $ .*03




52/69

Session 1.52

(ro'erties of Standard De,iation

It is the most widel" $sed meas$re ofdis'ersion* %Che!"che,s Ine7$alit"&

It is !ased on all the items and is rigidl"

defined* It is $sed to test the relia!ilit" of meas$res

calc$lated from sam'les* The standard de,iation is sensiti,e to the

'resence of e)treme ,al$es* It is not eas" to calc$late !" hand %$nli.e the

range&*



53/69

Session 1.53

*he(yshev=s 'ule

It 'ermits $s to ma.e statements a!o$tthe 'ercentage of o!ser,ations that

m$st !e within a s'ecified n$m!er of

standard de,iation from the mean

The 'ro'ortion of an" distri!$tion thatlies within k standard de,iations of the

mean is at least 898;k '


54/69

Session 1.54

For an" data set with mean %µ& andstandard de,iation %SD& the following

statements a''l"= At least of the o!ser,ations arewithin 5SD of its mean*

At least JJ* of the o!ser,ations arewithin 6SD of its mean*

*he(yshev=s 'ule



55/69

Session 1.55

Illustration

At "east 75

At least of the o!ser,ationsare within 5SD of its mean*



56/69

Session 1.56

Example

The midterm e)am scores of 4@@ STAT 4 st$dents

last semester had a mean of K and a standard

de,iation of J 'oints*

Applyin& the "hebyshe4s #+le! we can say that:

8. At least =)2 of the st+dents had scores

between >1 and ?8.

'. At least ??.12 of the st+dents had scoresbetween >8 and ?1.



57/69

Session 1.57

Coefficient of 0ariation %C0&

meas$re of relati,e ,ariation

$s$all" e)'ressed in 'ercent

shows ,ariation relati,e to mean $sed to com'are 5 or more gro$'s

Form$la =

00>×

=Mean

SC!



58/69

Session 1.58

Com'aring C0s

Stoc. A= A,erage (rice G (@

SD G (

C0 G 4@

Stoc. M= A,erage (rice G (4@@

SD G ( C0 G



59/69

Session 1.59

!easure of Se)ness

Descri!es the degree of de'art$res of thedistri!$tion of the data from s"mmetr"*

The degree of s.ewness is meas$red !"the coefficient of s.ewness denoted as S@ and com'$ted as

( )SD Median Mean K −=3S



60/69

Session 1.60

?hat is Symmetry@

A distrib+tion is said to besymmetric abo+t the mean!

if the distrib+tion to the left ofmean is the mirror ima&eBof the distrib+tion to the ri&htof the mean. $ikewise! asymmetric distrib+tion has

S@C( since its mean ise3+al to its median and itsmode.



61/69

Session 1.61

S@ @

positiely skewed

!easure of Se)ness

S@ @

ne&atiely skewed



62/69

Session 1.62

!easure of Aurtosis

Descri!es the e)tent of 'ea.edness orflatness of the distri!$tion of the data*

3eas$red !" coefficient of .$rtosis %@ &com'$ted as

( ) 4

1

4 3

N

i

i

X

K N

µ

σ

=

−

= −

∑



63/69

Session 1.63

K & 0

mesokurtic

K 0

leptokurtic

K * 0

platykurtic

!easure of Aurtosis



64/69

Session 1.64

Box


65/69

Session 1.65

The diagram is made $' of a bo7 which

lies !etween the first and third

7$artiles*The whiskers are the straight line

e)tending from the ends of the !o) to

the smallest and largest ,al$es that

are not o$tliers*

Box


66/69

Session 1.66

Steps to *onstruct a Box


67/69

Session 1.67

Step ': (lace mar.s at distances 4* *6# from

either end of the !o)* 4* *6# C8)<

100

+1 +3,

75 78 8560

1.5 IQR 1.5 IQR



68/69

Session 1.68

Ste' 6=Draw the hori#ontal line segments

.nown as the ;whis.ers< from each of the

end !o) to the largest and smallest ,al$es

in the data set that are not o$tliers*

An obseration beyond ±8.) *6# is ano+tlier.<

If the largest and smallest ,al$es in thedata set are o$tliers e)tend whis.ers $ntil4* *6# from either ends of the !o)*



69/69

Step >: For e,er" o$tlier draw a dot* If two or more dots

ha,e the same ,al$es draw the dots side !" side*

+1

+3

,

75 78 8560 100

1.5 IQR 1.5 IQR

9855

**


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