Measures of Position and Variability

 

MEASURES OF POSITION Von Christopher G. Chua, LPT, MST

 

Session Objectives In this session, graduate students enrolled in Statistical Methods are expected to acquire the ollo!ing co"petencies

#$ %esc %escri ri&e &e th the e thr three ee t'pes o quantiles$ ($ Inte Interp rpre rett the the pos posit itio ion n o a gi)en score &ased on its quantile ran*$

+$ o"p o"put ute e or or a spec specii-c c quartile, or decile o a gi)en ra! set o data and the percentile o a grouped data$

 This slidesho! presentation is "ade a)aila&le through the course !e&site, mathbychua.weebly.com %o!nload a cop' or 'our re)ie!ing purposes$

 

uic! "eview

.hat are "easures o central tendenc'/ .hat are the three co""onl' used "easures o central tendenc'/ %e-ne each$ .hen is it "ost appropriate appropriate to use each "easure/ .hen is it not

 

a

ro riate/

uic! "eview  The "edian is the "iddle"ost score in a distri&ution$  This i"plies that that hal or 01 percent o the scores lie &elo! 3or less than4 the "edian !hile the other hal o the scores lie a&o)e it$

6ighest Score

upper 012

M#$%& '

lo!er 012

5o!est Score

  a    t   a    d   r   o    3   n   o    i    t   u    &    i   r    t   4   s   t    i   e    %  s

 

The Me(ian)s Other Si*ni+cance

 The "edian is "ore "ore than 7ust a "easure o centrall tendenc' centra t endenc'$$ It also i"plies the location o a scor score e !ith respect to the other scores$ Such scores are called MEASURES OF

6ighest Score

upper 012

M#$%& '

lo!er 012

5o!est Score

  a    t   a    d   r   o    3   n   o    i    t   u    &    i   r    t   4   s   t    i   e    %  s

 

POSITION$ Consi(er this  8essie and 8essa are are identical t!ins !ho !ho go to the sa"e school and are in the sa"e grade le)el &ut &elong to di9erent sections$ On card distri&ution da', their "other noted that 8essie got a grade o :0 in Math !hile  8essa got :1 in the sa"e sa"e su&7ect$ oncerned, oncerned, she tal*ed to 8essa;s ad)iser regarding this$  The ad)iser said 8essa 8essa actuall' has the &etter grade$ 6o! is this possi&le/

6o! is this possi&le/  

Consi(er this  Ta*e a loo*  Ta*e loo* at the grades o&tained o&tained &' 8essie and 8essa in co"parison !ith the grades o their class"ates$  8essie;s class< :0 :1 :# :###th :+  

:0 :=

:> >1 >( >( >+

:: :> :>

:>

:1  8essa;s class< =th  =0 =0 =? == => => => => :1 :# :# :# :( :( :0

 

Measure o- Position (e+ne( Measures o position, also *no!n as Measures quantiles, di)ide the distri&ution, arranged in descending or ascending order, into se)eral equal parts depending on t'pe$  The quartiles di)ide di)ide the distri&ution into into our equal parts$  The deciles di)ide the distri&ution distri&ution into ten

equal parts$  

uartiles 6S

 The di)ide the @   quartiles di)ide

(02

distri&ution into our equal parts$

 The "iddle quartile is the "edian itsel$

  a    t   a    d   r (02 =02   o    3 M#$%&   n   o '    i    t   u (02 =02    &    i   r    t   4   s   t    i   e    %  s

 

 There are are three quartiles< quartiles< or lo!er quartile, or the "iddle quartile, and or upper quartile$

(02 5S

 

uartiles %eter"ine the )alue o the three quartiles in +> (# the ra! data< (1C#(( (: +>C++> "edian #: #: #> (1 (( (0 (? (: +1 +0  There are #0 "iddle"ost score is the :th$ +> +> +>scores$ BB The B0$"iddle"ost

Start !ith the "edian or "iddle quartile$  There are eight eight scores scores lesser or equal equal to the "edian$ The "iddle"ost o these scores is the lo!er quartile$  There are eight eight scores scores greater greater or equal to the

 

"edian$ The "iddle"ost o these scores is the

uartiles %eter"ine the )alue o the three quartiles in the =($0 ra! data< ?= =( =+ >1 "edian C# C+ =0 =0 0= 0>are?B ?B ?=The?> =1 is=# =( =+  There #: scores$ #: "edian "edian &et!een =( and =+$ =: >1 >B >= >> C( >>is =($0  There are > scores lesser or equal to the "edian$ C# is is ?=  There are > scores greater greater or equal to the "edian$ C+ is >1

 

$eciles 6S #1 2 #1 2

 The di)ide the distri&ution @   nine deciles di)ide into ten equal parts$

 The -th decile decile is the "edian "edian and is also equal to the "iddle quartile$ .hen !e consider , !e note that :1 percent perce nt o the distri&ution lies &elo! this )alue !hile (12 lies a&o)e it$ Si"ilarl', +1 percent o the scores are less than and the re"aining =1 percent are greater$

 

  a    t   a #1 2    d #1   r   o 2    3 M#$%&  n #1   o 2    i '    t #1   u    & 2    i   r #1    t   4   s   t 2    i   e    %  s #1

5S

2 #1 2 #1 2

 

$eciles the )alue o and in the ra! @%eter"ine     data<

? ? 0 ? ? ? ## #( ((#0   (1

? = = #0 #0

 

=

> #1 += +>  

 There are +1 scores$ %i)iding get +$ #: #> (1 (( this (B&' #1, (= !e(= (: +1 +0 +? "eans += that +> there B0 should B# &e three scores  This scores in &et!een e)er' decile$  The locations o the three deciles !e !ant to deter"ine deter"ine are gi)en$

 

Percentiles 6S

@D'  di)iding an' distri&ution into #11

#+ 2  

equal parts, there exists >> di9erent percentiles$

 The 01th percentile is the "edian$ Each quartile and decile is equi)alent to a percentile$ For exa"ple, the lo!er quartile is also the (0th percentile !hile the Bth decile is the sa"e as $  The :=th percentile accounts or the score !herein :=2 o the data alls

 

  a    t   a :=    d   r 2   o    3 M#$%&  n   o    i '    t   u    &    i   r    t   4   s   t    i   e    %  s

5S

 

&elo! it !hile the #+2 is a&o)e it$

Percentiles Groupe( $ata/ Percentiles are used or data containing a relati)el' Percentiles large sa"ple or population sie$ So"eti"es, ra! scores are trans"uted into its corresponding percentile$ percentile$ This nu"&er is called the percentile ran*$  The or"ula used or percentile percentile is< requenc'  cu"ulati)e sa"ple sie ti"es the

&eore the class

 )alue o *

Percentile score

class sie

  lo!er &oundar' o the class

 requenc' o the class

 

Percentiles Groupe( $ata/ or the )alue o the (=   @Sol)e   percentile in the gi)en grouped data$ th

 

Identi' the location o the (=   percentile -rst &' co"puting or the )alue o $ th

(=th  percentile class

5ocate this )alue &ased on the cu"ulati)e requencies to deter"ine the percentile class$

Class %nterval s 0  ((

0

0

(+  B1

>

#B

B#  0:

#B

(:

0>  =?

#1

+:

==  >B >0  ##(

0 (

B+ B0

All )alues needed in the or"ula "ust  

Percentiles Groupe( $ata/ Sol)e or the )alue o the (= th  percentile in the gi)en grouped data$  

(=th  percentile class

 This "eans that (= percent percent o the scores all &elo! +?$: !hile =+ percent are a&o)e it$

Class %nterval s 0  ((

0

0

(+  B1

>

#B

B#  0:

#B

(:

0>  =?

#1

+:

==  >B >0  ##(

0 (

B+ B0

 

Percentiles Groupe( $ata/ o"pute or the )alue o the ?: th  percentile in the gi)en grouped data$  

?:  percentile class

Class %nterval s 0  ((

0

0

(+  B1

>

#B

B#  0:

#B

(:

0>  =?

#1

+:

==  >B >0  ##(

0 (

B+ B0

th

?: percent o the scores are less than

?+$=$  

MEASURES OF GARIADI5ITH

 

Session Objectives In this session, graduate students enrolled in Statistical Methods are expected to acquire the ollo!ing co"petencies

#$ Expl Explain ain th the e co conc ncep eptt o o )aria&ilit' as de-ned in statistics$ ($ o"p o"par are e th the e "o "ost st co""on "easures o )aria&ilit' &' stating the ad)antages and disad)antages o using

+$ o"p o"put ute e or or th the e ran range ge,, )ariance, and standard de)iation o a gi)en set o ra! or grouped data$

 This slidesho! presentation is "ade a)aila&le through the course each$ !e&site, mathbychua.weebly.com

%o!nload a cop' or 'our re)ie!ing purposes$  

Loo! bac! an( Learn  Ten heads o a a"il' each  Ten each ro" t!o &aranga's in the sa"e "unicipalit' ha)e &een as*ed to state ho! "uch the' earn per da'$ The data is gi)en &elo!$ Drg' A<

##1

#>1

(11

(B1 (>1

+01 +01

+01 +?1 ##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 o"pute or the "ean and "edian o &oth data sets$

 

 The degree degree to !hich nu"erical data tend to spr spread ead a&out an a)erage )alue is called the (ispersion, or variation, o the data$ Cuantities that ai" to represent such characteristic are called measures o- variability$

 

 The "ost co""on "easures "easures o )aria&ilit' or dispersion are the range, interquartile range, )ariance, and standard de)iation$

 

The "an*e Pros  Si"plest "easure o )aria&ilit'

Drg' A< ##1 #>1 (11 (B1 (>1 (>1 +01 +01 +01 +?1 ##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 In the case o the t!o sets o data gi)en a&o)e, the "easures o central tendenc' are equal$ 6o!e)er, &' loo*ing into the indi)idual scores, !e can i"pl' that there is a relati)el' large di9erence &et!een the extre"e"ost scores in each data set$  The range is the di9erence di9erence &et!een the highest and the lo!est scores$





Eas' to co"pute$ Expressed in the sa"e unit as the ra! scores$

Cons    %oes not ta*e into account all

scores in the

 

The %nter0uartile "an*e  The Interquartile Range Range 3ICR4 is a "easure that @   indicates the extent to !hich the central 012 o )alues !ithin the dataset are a re dispersed$ ICR is the di9erence &et!een the upper and lo!er quartiles$ That is,

Drg' A< ##1 #>1 (11 (B1 (>1 (>1 +01 +01 +01 +?1 ##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 o"pute or and co"pare the ICRs o the t!o



Pros  Eas' to co"pute$ Expressed in the sa"e unit as the ra! scores$  Not easil' a9ected &' outliers$ Cons    %oes not ta*e into

data sets$

account all

 

The Variance  The co"putation o )ariance di9ers @   depending on !hether the data is ta*en ro" a population or ro" a sa"ple$

Population variance o a set o data is the su" o the squares o the di9erences &et!een each o&ser)ation in the sa"ple and the sa"ple "ean di)ided &' the population sie$ Sample variance is si"ilar except that # is su&tracted ro" the sa"ple sie$ In s'"&ols,  

Pros  Not easil' a9ected &' outliers$  Ta*es a*es into   T account all scores in the distri&ution$ Cons  Not easil' co"puted co"pared to the other "easures o

)aria&ilit'$  

The Variance or the )ariance o the gi)en data$ @o"pute   Drg' A< ##1 #>1 (11 (B1 (>1 (>1 +01

+01 +01

+01 +?1 ##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 o"puting o data ro" Drg' A<

Since the data is ta*en ro" a sa"ple, !e use the or"ula, D' co"putation, +$

##1

J(0+

?B 11>

#>1

J#=+

(> >(>

(11

J#?+

(? 0?>

(B1

J#(+

#0 #(>

(>1

J =+

0 +(>

+01

J #+

#?>

+01

J #+

#?>

+01

J #+

#?>

+?1

J+

>

##>1

:(= ?:+ >(> :(0

B#1  

The Variance or the )ariance o the data ro" Drg'$ @o"pute   D$ Drg' A< ##1 #>1 (11 (B1 (>1 (>1 +01 +01 +01 +01 +?1 ##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 Since the data is ta*en ro" a sa"ple, !e use the or"ula, D' co"putation, +$

(>1

J =+

0+(>

(>1

J =+

0+(>

(>1 + +1 11 1 + +( (1 1

J =+ JJ? ?+ + JJB B+ +

0+(> + +> >? ?> > # #: :B B> >

+ +? ?1 1 B11 B11 B01 B01 B?1 B?1 B=1 B=1

JJ+ + += += := := >= >= #1= #1=

> > #+?> #+?> =0?> =0?> >B1> >B1> ##BB> ##BB>

K

0# 0 # ?#1 ?#1

 

The Stan(ar( $eviation Pros  Not easil' a9ected &' outliers$  Ta*es a*es into   T account all scores in the distri&ution$

 The Stan(ar( $eviation o a gi)en set o @   data is the square root o the )ariance$  The co"putation o standard de)iation depends on the )ariance$ Thereore, the or"ula or population standard de)iation is di9erent ro" the sa"ple standard de)iation$ 

Expressed in the sa"e unit as the ra! scores$

Cons  Not easil'

co"puted  

The Stan(ar( $eviation the sa"e data sets, @Li)en   Drg' A< ##1 #>1 (11

(B1 (>1 +01 +01 +01 +01 +?1

##>1 Drg' D< (>1 (>1 (>1 +11 +(1 +?1 B11 B01 B?1 B=1 And ha)ing o"puted or the )ariance o the distri&utions, !e

get the ollo!ing )alues or the standard de)iation< Drg'$ A< Drg' D< D' co"parison, the data on dail' inco"e o the heads o a"ilies in Drg' A is "ore dispersed than that in Drg' D$

 

The Stan(ar( $eviation o"pute or the )ariance and the standard de)iation o the gi)en distri&ution$

+0 +> B1 BB B0 B0 B= B: 0+

0B  

Variance an( Stan(ar( $eviation -or Groupe( $ata

@I the   data is grouped, use the or"ula< Class %ntervals 0  ((

0

(+  B1

>

B#  0:

#B

0>  =?

#1

==  >B

0

>0  ##(

(

 

Variance an( Stan(ar( $eviation -or Groupe( $ata Class %ntervals

0  ((

0

#+$0

?=$0

(+  B1

>

+#$0

(:+$0

J(1$:

B#  0:

#B

B> >$$0 0 B

?> >+ + ?

($$: : JJ(

=$$: :B B =

#1 1> >$$= =? ? #

0> > = =? ? 0

#1 1 #

?= =$$0 0 ?

?= =0 0 ?

#0 0$$( ( #

(+ +# #$$1 1B B (

( (+ +# #1 1$$B B

==  >B > 0 # ## #( ( >0

0 ( (

:0$0 # 1 +$$0 0 #1+

B(=$0 (1 1= = ( ( +0+$0 +0+$0 (

J+:$: #010$BB

=0(=$(

B+($?B +:>+$=?

++$( ##1($(B 00##$( 0 #$$( ( ( (? ?( (# #$$B BB B 0 0( (B B( ($$: :: : 0# (B (B 0>0$( 0>0$(

 

 1our turn  1our o"pute or the sa"ple )ariance and sa"ple standard de)iation o the gi)en grouped distri&ution$ Class %ntervals

BJ#B

=

#0J(0

#1

(?J+?

((

+=JB=

##

B:J0:

0

 

%mportant "emin(ers an( Concerns %o!nload 5earning Tas* + ro" theon course !e&site  8anuar' #+ #+ and su&"it on  8anuar' (# (# during the Midter" Exa"$ Su&"it all other 5earning tas*s on

Covera*e o- the mi((le o- the term e2amination3 Dasic Statistical Ter"s Sa"pling and its Techniques Nature o %ata Methods o %ata Presentation Lrouping %ata< Frequenc' %istri&ution Ta&les, 6istogra"s, Frequenc' Poli'gons, and Plots Su""ation Measures o entral Tendenc' Measures o Position Measures o Garia&ilit'

Type o- test3 Modi-ed true or alse, o"putation and anal'sis 4rin* calculators an(5or laptops. %t will be an 6open notes7 test.

the sa"e date$  

T8&'9  S: