Measures of Position and Variability
October 13, 2022 | Author: Anonymous | Category: N/A
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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 sie$ 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 sie ti"es the
&eore the class
)alue o *
Percentile score
class sie
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 sie$ Sample variance is si"ilar except that # is su&tracted ro" the sa"ple sie$ 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$ Thereore, 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:
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