central tendency 29 mac 2016

8/18/2019 Central Tendency 29 Mac 2016

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SCOPE• MODE

• MEDIAN

• STANDARD DEVIATION

• INTREPRETATION OF DATA

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Mean, Median, Mode (Measures of Central Tendency)

A measure o !e"#ra$ #e"de"!% &s a" a'erage 'a$ue &" a se#

o s!ores.

T(ere are #(ree #%)es o measures o !e"#ra$ #e"de"!%*

mea"+

med&a" a"d

mode.

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MeanMean &s #(e mos# !ommo"$% used measure o !e"#ra$#e"de"!%.

-(a# &s mea"

EAMP0E *

add&"g a$$ o #(e s!ores &" a se# a"d d&'&d&"g #(&s sum %#(e "umer o s!ores &s !a$$ed mean. For eam)$e+ #(e

mea" or s!ores 73+ 45 a"d 133 &" a #es# &s*

7345133 6 45

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Median&s #(e m&dd$e s!ore 8(e" #(e #o#a$ "umer o s!ores &s odd

or

#(e a'erage o #(e #8o m&dd$e s!ores & #(e #o#a$ "umer o

s!ores &s e'e" &" a" as!e"d&"g order o d&s#r&u#&o".

Eam)$e 1* 9as!e"d&"g order:

SCORES * ,5+ 3+ 5;+ 73+


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Mode &s #(e s!ore #(a# (as #(e (&g(es# re=ue"!% &" a

d&s#r&u#&o". A d&s#r&u#&o" ma% (a'e o"e or more modes.

>o8e'er+ ma% "o# (a'e a mode a# a$$.

Eam)$e 1* 7;+ 55+ ,+ 43+ ;3+


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Which set of data has ONE MODE?

9, 11, 16, 8, 16

9, 12, 16, 9, 7, 17, 18

14, 8, 10, 7, 18

A

C

B

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Which set of data has NO MODE?

13, 12, 12, 11, 12

9, 11, 16, 6, 7, 17, 18

18, 7, 10, 7, 18

A

C

B

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Which set of data has

MORE THAN ONE MODE?

9, 11, 16, 8, 16

9, 11, 16, 6, 7, 17, 18

18, 7, 10, 7, 18

A

C

B

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F&"d #(e?.

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Mode

The mode is the most frequent or popular score in the population. This is usually evident

during the drawing of frequency tables. It is not frequently used as the median and mean in

the classroom because :

Median

This is the middle score after all the scores have been arranged in order of magnitude i.e.

50% of the score are on either side of it. What is the drawbac of using median :

Mea"T(&s &s #(e a'erage o a$$ #(e s!ores a"d &s o#a&"ed % add&"g #(e s!ores

#oge#(er a"d d&'&d&"g #(e sum % #(e "umer o s!ores. Is useu$

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1.T(ere ma% e more #(a" o"e mode2.T(ere ma% e "o mode a# a$$ & "o"e o #(e da#a &s #(e same

,.I# ma% "o# a!!ura#e$% re)rese"# #(e da#a

Ver% $arge or 'er% sma$$ "umers !a" d&s#or# #(e a"s8er. >o8e'er &s s#&$$ more

a!!ura#e #(a" med&a" a"d mode &" #erms o !a$!u$a#&o" o" d&s#r&u#&o" o s!ores .

Ta@es a $o"g #&me #o !a$!u$a#e or a 'er% $arge se# o da#a


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If in a class of nine students :

the scores are !"# $5# $# '0# &5# !0# (0 and 5.

)ind the

a*mean.

b*mode

c*median

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Standard Deviation

es&des @"o8&"g #(e measures o !e"#ra$ #e"de"!%+ o"e

(as #o @"o8 (o8 s)read ou# #(e s!ores are. O"e 8a% &s #o

de#erm&"e #(e s#a"dard de'&a#&o".

S#a"dard De'&a#&o"+ σ 6 N

X X ∑ − +

!*,

Interpretation of Standard Deviation Values

T(e sma$$er #(e 'a$ue o s#a"dard de'&a#&o"+ #(e sma$$er #(e

s)read o #(e d&s#r&u#&o" a"d #(&s &m)$&es #(e da#a are !$oser

#o ea!( o#(er 9(omoge"eous:B #(e &gger #(e s.d.+ #(e more

(e#eroge"eous #(e da#a &s.

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Co"s&der #(e o$$o8&"g eam)$e or #(e !a$!u$a#&o" o

s#a"dard de'&a#&o" or #(e o$$o8&"g da#a*

T(e s!ore o 5 s#ude"#s are 1+ 2 + ,+ a"d 5.

Mea"+ 6 15/5 6 ,

N

X X ∑ − !*,S#a"dard De'&a#&o"+ σ 6

S!ore 9: 9 : 9 :2

5 2

,2

1

N 6 5 Σ 9 :2 6

6

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Co"s&der #(e o$$o8&"g eam)$e or #(e !a$!u$a#&o" o

s#a"dard de'&a#&o" or #(e o$$o8&"g da#a*

T(e s!ore o 5 s#ude"#s are 1+ 2 + ,+ a"d 5.

Mea"+ 6 15/5 6 ,

N

X X ∑ − !*,S#a"dard De'&a#&o"+ σ 6 6 1.1

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Score () ! ( " )

( " )#

5 2

1 1, 3 3

2 1 1

1 2

N65 13

Σ 9 :2 6


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Students

Marks

obtained(x)

A 20

B 45

C 39

D 40

E 42

F 48

G 30

H 46

I 41

total

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Da#a !a" e d&s#r&u#ed 9s)read ou#: &" d&ere"# 8a%s

I# !a" e s)read ou#

more o" #(e $e#

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Or more o" #(e r&g(#

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$ormal Distri%ution Curve

A Norma$ D&s#r&u#&o" Cur'e &s a e$$s(a)ed !ur'e 8(e" #(e s!ores are

"orma$$% d&s#r&u#ed. T(e area u"der #(e !ur'e re)rese"#s a$$ 9133: o

#(e s!ores a"d 53 o #(e s!ores are ao'e #(e mea" a"d 53 o #(es!ores are e$o8 #(e mea".

F&gure 4* A Norma$ D&s#r&u#&o" Cur'e

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I# # $ # d #( # #( $


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I# mus# a$so e "o#ed #(a# #(e 'a$ue

or #(e mea"+ #(e med&a"+ a"d #(e

mode are #(e same. Mos# s!ores

are "ear #(e mea" a"d #(e ur#(er

rom #(e mea" a s!ore &s #(e e8er

#(e "umer o )erso"s 8(o a##a&"#(a# s!ore.

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A e$$ !ur'e o$$o8s #(e ;4


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Eam)$e*


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Standard Scores (&"score and T"score)

A s#a"dard s!ore e)resses #(e )os&o" o a" &"d&'&dua$

s!ore &" #erms o (o8 ma"% s#a"dard de'&a#&o"s #(e s!ore

&s ao'e or e$o8 #(e mea" o #(e d&s#r&u#&o". I# &s usua$$%

re)rese"#ed % &"score or T"score'

&"score &s !om)u#ed % us&"g #(e o$$o8&"g ormu$a*

σ

*, X X −G 6

T"score &s !om)u#ed % us&"g #(e o$$o8&"g ormu$a*

T 6 53 13Gdsg/21.7.15 ,3


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Eam)$e*

T(e s!ores or I"#a"+ D(ara"&+ >aGru$+ Na&du a"d Co"rad &"

a #es# are 5+ 4+ 13+ a"d , res)e!#&'e$%. F&"d #(e Gs!orea"d Ts!ore or >aGru$.

X 6 95 4 13 ,:/5 6 ; S#d. de'. 6 2.;1

Gs!ore 6 913;:/2.;1 6 1.5

Ts!ore 6 53 13G 6 53 1391.5: 6 ;5.

σ

*, X X −G 6

T 6 53 13G

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A H s!ore 9a$so @"o8" as a G 'a$ue+ s#a"dard s!ore+ a"d "orma$ s!ore: &s used #o

des!r&e a )ar#&!u$ar s!ore &" #erms o 8(ere s &"#o a" o'era$$ grou) o s!ores. I"

o#(er 8ords+ a H s!ore &s a" ord&"ar% s!ore #ra"sormed so #(a# e##er des!r&es #(e

$o!a#&o" o #(a# s!ore &" a d&s#r&u#&o".

A H s!ore (as a mea" o Gero a"d a s#a"dard de'&a#&o" o o"e.

T S!ores

T s!ores are used #o #e$$ &"d&'&dua$s (o8 ar #(e&r s!ore &s rom #(e mea". T s!ores

(a'e a mea" o 53 a"d a s#a"dard de'&a#&o" o 13.

T(ereore+ & a s#ude"#s ra8 s!ore 8as !o"'er#ed #o a T s!ore a"d #(e&r T s!ore 8as

73 8ou$d &" #ur" mea" #(a# #(e&r s!ore 8as 23 )o&"#s ao'e #(e mea".

O"e ad'a"#age o us&"g a T s!ore o'er a G s!ore &s #(a# T s!ores are re$a#&'e$% eas% #o

e)$a&" #o )are"#s 8(e" re)or#&"g #(e s#ude"#s assessme"# s!ores.

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-( # d d& I# ( $ # @ d & &


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Eam)$e* Ms Sa$$% &s mar@&"g a #es#.

>ere are #(e s#ude"#s resu$#s 9ou# o ;3 )o&"#s:*

23+ 15+ 2;+ ,2+ 14+ 24+ ,5+ 1+ 2;+ 22+ 17

Mos# s#ude"#s d&d"J# e'e" ge# ,3 ou# o ;3+ a"d mos# 8&$$ a&$.

T(e #es# mus# (a'e ee" rea$$% (ard+ so #(e Ms Sa$$% de!&des #o S#a"dard&Ge a$$

#(e s!ores a"d o"$% a&$ )eo)$e?.. 1 s#a"dard de'&a#&o" e$o8 #(e mea".

T(e Mea" &s 2,+ a"d #(e S#a"dard De'&a#&o" &s ;.;+ a"d #(ese are #(e S#a"dard

S!ores*

3.5+ 1.21+ 3.5+ 1.,;+ 3.7;+ 3.7;+ 1.42+ 1.,;+ 3.5+ 3.15+ 3.


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O"$% 2 s#ude"#s 8&$$ a&$ 9#(e o"es 8(o s!ored 15 a"d 1 o" #(e #es#:

#

#'*+*'

scores Standardscores

23 3.5

15 1.21

2; 3.5

,2 1.,;

14 3.7;

24 3.7;

,5 1.42

1 1.,;

2; 3.5

22 3.15

17 3.


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