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Coverage Measures of Central Tendency

Mean Median Mode

Measures of Variability and Dispersion Range

Average deviation Variance Standard deviation

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Introduction to Notations

If variable X is the variable ofinterest and that n

!easure!ents are ta"en#

then the notation X$ X % X & ' Xn (ill be used to represent nobservations )

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Sig!a

Indicates * su!!ation of +

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Su!!ation Notation

If variable X is the variable of interestand that n !easure!ents are ta"en#

the su! of n observations can be(ritten as

X i = X 1+X 2+ +X nn

i=1

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Su!!ation Notation

X i = X 1+X 2+ +X nn

i=1

,pper li!it of su!!ation

-o(er li!it of su!!ation

.ree" letter Sigma

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Rules of Su!!ation

su!!ation of thesu! of variables

is

(X i+Y i) = X i+ Y i n

i=1

n

i=1

n

i=1

the su! oftheir su!!ations

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(a i+b i++z i)

= a i+ b i + + z i

n

i=1

n

i=1

n

i=1

n

i=1

The su!!ation of the su! of

variables is'

the su! of their su!!atio

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Rules of Su!!ation

cX i = c X i

= c(X 1+X 2+ +X n)

n

i=1

n

i=1

If c is a constant then'

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Rules of Su!!ation

c = ncn

i=1

The su!!ation of a constant isthe product of upper li!it ofsu!!ation n and constant c )

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MEASURES OF CENTRALTENDENCY

Statistics in Research

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Mean The su! of all values of the

observations divided by the totalnu!ber of observations

The su! of all scores divided by thetotal fre/uency

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Population mean = X i N

i=1

NSample mean x = X i

n

i=1

n

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f iX in

i=1

n

Mean in an ,ngrouped0re/uency

= (f 1X1+f 2X2+ +f nXn)

(here f is the fre/uency

of the occurring score

n

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1roperties 2 Mean The !ost stable !easure of central

tendency Can be a3ected by e4tre!e values Its value !ay not be an actual value in

the data set If a constant c is added5substracted to

all values the ne( !ean (illincrease5decrease by the sa!e a!ountc

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Median 1ositional !iddle of an array of data Divides ran"ed values into halves

(ith 678 larger than and 678s!aller than the !edian value)

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If n is odd 9

M = X (n+1)!2If n is even 9

M = X n!2 + X (n!2)+1

2

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1roperties 2 Median The !edian is a positional !easure Can be deter!ined only if arranged in

order Its value !ay not be an actual value in

the data set It is a3ected by the position of ite!s in

the series but not by the value of eachite!

A3ected less by e4tre!e values

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Mode Value that occurs !ost fre/uently in

the data set -ocates the point (here scores occur

(ith the greatest density -ess popular co!pared to !ean and

!edian !easures

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1roperties 2 Mode It !ay not e4ist or if it does it !ay

not be uni/ue Not a3ected by e4tre!e values Applicable for both /ualitative and

/uantitative data

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Range Measure of distance along the

nu!ber line over (here data e4ists :4clusive and inclusive range

:4clusive range ; largest score 2s!allest score

Inclusive range ; upper li!it 2 lo(erli!it

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1roperties 2 Range Rough and general !easure of

dispersion -argest and s!allest e4tre!e values

deter!ine the range Does not describe distribution of

values (ithin the upper and lo(ere4tre!es

Does not depend on nu!ber of data

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M" = # Xi $ X #

n

i=1

n

M%" = # Xi $ M #n

i=1

n

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1roperties > AbsoluteDeviation

Measures variability of values in thedata set

Indicates ho( co!pact the group ison a certain !easure

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Variance Average of the s/uare of deviations

!easured fro! the !ean 1opulation variance

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?%

= ( X i $ )2

N

i=1

N

s % = ( X i & X )2n

i=1

n $1

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s % = n X i2 $ ( X i )2n

i=1

n(n $1)

n

i=1

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1roperties > Variance Addition5subtraction of a constant c

to each score (ill not change thevariance of the scores

Multiplying each score by a constantc changes the variance resulting in ane( variance !ultiplied by c %

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Standard Deviation S/uare root of the average of the

s/uare of deviations !easured fro!the !ean > s/uare root of thevariance

1opulation standard deviation

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? = ( X i $ )2

N

i=1

N

s = ( X i & X )

2n

i=1

n $1

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@hy n2$ Degrees of freedo!

Measure of ho( !uch precision anesti!ate of variation has

.eneral rule is that the degrees offreedo! decrease as !ore para!etershave to be esti!ated

Xbar esti!ates ,sing an esti!ated !ean to nd the

standard deviation causes the loss of EN:degree of freedo!

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1roperties > StandardDeviation

Most used !easure of variability A3ected by every value of every

observation -ess a3ected by Fuctuations and

e4tre!e values

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1roperties > StandardDeviation

Addition5subtraction of a constant cto each score (ill not change thestandard of the scores

Multiplying each score by a constantc changes the standard deviationresulting in a ne( standard deviation!ultiplied by c

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Choosing a !easure Range

Data are too little or scattered to Gustify!ore precise and laborious !easures

Need to "no( only the total spread of scores Absolute Deviation

0ind and (eigh deviations fro! the!ean5!edian

:4tre!e values unduly s"e(s the standarddeviation

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Choosing a !easure Standard Deviation

Need a !easure (ith the best stability :3ect of e4tre!e values have been

dee!ed acceptable Co!pare and correlate (ith other data

sets

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FREQUENCYDISTRIBUTION

Statistics in Research

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H HJ KJ H% 6& HK K% L% L LH JKH% HJ KL H$ 67 H6 K7 L$ L LK J$H% HH KK KJ 67 H6 6J L7 L% L6 LL

H% HH KK KJ 67 H6 K7 L$ L& L6 LJH& HL KL H7 67 H6 K7 L$ L& LK LJH& 6J K6 KJ 67 H6 HH L7 L% L LH

H& HJ KL H$ 6$ HK K% L$ L LH J%H& HJ KL H$ 6% HK K% L% L LH JH HJ KL H$ 6& HK K% L% L LH J

67 6H K& KJ H% H HH L7 L% L LH

Ra( data

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67 6H K& KJ H% H HH L7 L% L LH67 6J K6 KJ H% H6 HH L7 L% L LH67 6J KK KJ H% H6 HH L7 L% L6 LL

67 K7 KK KJ H% H6 HH L$ L& L6 LJ67 K7 KL H7 H& H6 HL L$ L& LK LJ67 K7 KL H$ H& H6 HJ L$ L LK J$

6$ K% KL H$ H& HK HJ L$ L LH J%6% K% KL H$ H& HK HJ L% L LH J6& K% KL H$ H HK HJ L% L LH J

6& K% KJ H% H HK HJ L% L LH JK

Array

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0re/uency Distribution Table

Class 0re/uency Nu!ber of observations (ithin a class f

Class -i!its :nd nu!bers of the class

Class Interval Interval bet(een the upper and lo(er

class li!its ie9 X upper li!it X lo(er li!it

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Class Ooundaries True li!its of the class half(ay bet(een class

li!it of the current class and that of thepreceding5succeeding class -CO and ,CO

Class SiPe Di3erence bet(een ,CO and -CO

ie9 X ,CO 2 X-CO Class Mar"

Midpoint of the class interval average valueof the upper and lo(er class li!its ie) X upper li!it 2 Xlo(er li!it

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Constructing an 0DT Deter!ine nu!ber of classes

Sturges 0or!ula Q ; $ &)&%% log n S/uare Root Q ; s/rt

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Constructing an 0DT Deter!ine lo(er class li!it

-o(est class should not be e!pty !ustcontain the lo(est value in the data set

Deter!ine succeeding lo(er classli!its by adding class siPe C to thecurrent lo(er class li!it

Tally fre/uencies

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0 / i ib i

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Class Frequency LCB UCB RF

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Ether Ter!s Relative fre/uency R0

Class fre/uency divided by nu!ber ofobservations ie) R0 ; f i 5 n

Relative 0re/uency 1ercentage R01 R0 ;

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Mean fro! an 0D

X = f iX i'

i=1

f i'

i=1

(here X i ; class !ar" of the i th cl

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Median fro! an 0D

M = * M + n!2 $ , M $1

(here * M ; lo(er class boundary of !edianclass

, M $1 ; less than cu!ulative fre/uency

f M

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Mode fro! an 0D

Mo = * Mo + f Mo $ f Mo$1

(here * Mo ; lo(er class boundary of !odal classf Mo- f Mo$1- f Mo+1 ; fre/uency of !odal class class

preceding and class succeeding the

2f Mo $ f Mo$1 $ f Mo+1

M D i i f !

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MD = f i #Xi $ X#n

i=1

n

Mean Deviation fro! an0D

(here X i ; class !ar" of the i th class n ;total nu!ber of observations# total

fre/uency ie) n = f i

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s % = f i(X i $ X)2n

i=1

(n $1)

Variance fro! an 0D


fre/uency ie) n = f i

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s % = n f iX i2 $ ( f iX i )2n

i=1

n(n $1)

Variance fro! an 0D

n

i=1


fre/uency ie) n = f

statistics-1221151261161613-8

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