!!!!Lecture!4!

!

Summarization!of!Data:!Algebraic!Approach!

!

!

There!are!three!aspects!we!look!at!

!

1. Where!does!the!centre!of!the!data!lie?!!

!

Measures!of!Central!Tendency!

!

2. How!is!the!data!spread!across!the!

centre?!

!

Measures!of!Dispersion.!

!

3. Is!the!data!symmetric?!

!

Measures!of!skewness.!!

!

!

!

!

!

!

!!!Central!tendency!

!

â€¢ The!mean:!Given!a!numerical!data!set!

â‚¬

{x1,x2,x3,...xn}

,!the!Arithmetic!Mean!is!

given!by!

!

â‚¬

x

_

=x1+x2+...xn

n=1

n

xi

i=1

n

âˆ‘

!

!

!

Other!(rarely!used)!means!

!

â€¢ Geometric!mean!

!

G.M!=!

â‚¬

(x1*x2* ...xn)1 / n

!

!

â€¢ Harmonic!Mean!

!!!!H.M!=!

â‚¬

n

1

x1

+1

x2

+....+1

xn

!

!

Applications:!

!

(i) Average!interest!rate!

(ii) Average!Speed!

â€¢ The!median:!This!can!be!used!when!the!

data!is!ordinal.!The!median!is!defined!to!

be!the!value!of!Q2!such!that!half!the!

observations!are!below!Q2!and!half!are!

above!Q2!

!

If!we!ranked!the!data!as!x(1),!x(2),â€¦.!x(n),!

where!

!

! x(1)!â‰¤!x(2)!â‰¤!x(3)â‰¤â€¦â€¦.â‰¤x(n),!

!

then!if!n!is!odd,!then!the!median!is!

x({n+1}/2).!When!n!is!even,!then!the!median!

is!Â½[!x(n/2)!+x((n+1)/2)]!

!

!

â€¢ The!mode:!This!is!used!when!we!have!

grouped!or!frequency!data.!The!mode!is!

the!one!with!the!highest!frequency!

!

!

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