week 3-part 2.pdf
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Fall 2013
School
University of Waterloo
Department
Statistics
Course Code
STAT231
Professor
Matthias Schonlau
Lecture
3
!!!!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!
!
• The!mean:!Given!a!numerical!data!set!
€
{x1,x2,x3,...xn}
,!the!Arithmetic!Mean!is!
given!by!
!
€
x
_
=x1+x2+...xn
n=1
n
xi
i=1
n
∑
!
!
!
Other!(rarely!used)!means!
!
• Geometric!mean!
!
G.M!=!
€
(x1*x2* ...xn)1 / n
!
!
• Harmonic!Mean!
!!!!H.M!=!
€
n
1
x1
+1
x2
+....+1
xn
!
!
Applications:!
!
(i) Average!interest!rate!
(ii) Average!Speed!
• 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),….!x(n),!
where!
!
! x(1)!≤!x(2)!≤!x(3)≤…….≤x(n),!
!
then!if!n!is!odd,!then!the!median!is!
x({n+1}/2).!When!n!is!even,!then!the!median!
is!½[!x(n/2)!+x((n+1)/2)]!
!
!
• 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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