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# Measures of central tendency and dispersion Notes

## : Measures of central tendency and dispersion Notes

``` Page 1

Lecture 3: Measures of central tendency
and dispersion
Module 7
Page 2

Lecture 3: Measures of central tendency
and dispersion
Module 7
Measures of Central Tendency
Module 7
1
1
E(x) =  = ( ) ,  for parameter estimate
Arithmatic Mean ( ) =   .......for sample estimate
Most frequently occuring value
1)  0 & 0
2) Value of x associated
Mean:
Mo

de:
with
ma
a
a
µ
+
-
=
=
??
= <
??
?
?
n
i
i
n
i
x f x dx
xx
ff
xx
f ( )
The pdf takes the maximum value at the mode.
x
xi
x
Page 3

Lecture 3: Measures of central tendency
and dispersion
Module 7
Measures of Central Tendency
Module 7
1
1
E(x) =  = ( ) ,  for parameter estimate
Arithmatic Mean ( ) =   .......for sample estimate
Most frequently occuring value
1)  0 & 0
2) Value of x associated
Mean:
Mo

de:
with
ma
a
a
µ
+
-
=
=
??
= <
??
?
?
n
i
i
n
i
x f x dx
xx
ff
xx
f ( )
The pdf takes the maximum value at the mode.
x
xi
x
Measures of Central Tendency                            Contd…
Module 7
It divides the area under the pdf curve into two halves.
i.e Area is 50%
= ( ) = P[X ] = 0.5
:
[ It is the observation such that
half the values lie on either side of it ]
Median:
µ
µµ
-
=
?
med
med x med
d
n
P x dx
Def
(b)
Median
Median
(c)
Median
(a)
Page 4

Lecture 3: Measures of central tendency
and dispersion
Module 7
Measures of Central Tendency
Module 7
1
1
E(x) =  = ( ) ,  for parameter estimate
Arithmatic Mean ( ) =   .......for sample estimate
Most frequently occuring value
1)  0 & 0
2) Value of x associated
Mean:
Mo

de:
with
ma
a
a
µ
+
-
=
=
??
= <
??
?
?
n
i
i
n
i
x f x dx
xx
ff
xx
f ( )
The pdf takes the maximum value at the mode.
x
xi
x
Measures of Central Tendency                            Contd…
Module 7
It divides the area under the pdf curve into two halves.
i.e Area is 50%
= ( ) = P[X ] = 0.5
:
[ It is the observation such that
half the values lie on either side of it ]
Median:
µ
µµ
-
=
?
med
med x med
d
n
P x dx
Def
(b)
Median
Median
(c)
Median
(a)
[ ]
[ ] [ ]
[ ] [ ] [ ]
aa
aa
aa aa
aa aa
--
-- --
+= +
+ +
? += +
??
?? ??
L ,  E ( ) ( , )
=   ( , ) y ( , ) =E E

E E E

et X Y x y f x y dxdy
x f x y dxdy f x y dxdy X Y
XY X Y
[ ]
aa
aa
a
a
--
-
=
??
?
(X, Y) an independent RVs with a joint pdf of f(x,y)
E , ( , )
For independent variable f(x,y) = g(x)  ( ) ( x and y are indpendent)
=   (
X Y xy f x y dxdy
x h y because
xy g x
[ ] [ ]
[ ] [ ] [ ]
a aa
a aa - --
•+
? += ×
? ??
) ( ) = ( ) y h( ) = E E

E E   E  (If x and Y are independent)

h y dxdy x g x dx y dy X Y
XY X Y
Module 7
Measures of Central Tendency                            Contd…
Page 5

Lecture 3: Measures of central tendency
and dispersion
Module 7
Measures of Central Tendency
Module 7
1
1
E(x) =  = ( ) ,  for parameter estimate
Arithmatic Mean ( ) =   .......for sample estimate
Most frequently occuring value
1)  0 & 0
2) Value of x associated
Mean:
Mo

de:
with
ma
a
a
µ
+
-
=
=
??
= <
??
?
?
n
i
i
n
i
x f x dx
xx
ff
xx
f ( )
The pdf takes the maximum value at the mode.
x
xi
x
Measures of Central Tendency                            Contd…
Module 7
It divides the area under the pdf curve into two halves.
i.e Area is 50%
= ( ) = P[X ] = 0.5
:
[ It is the observation such that
half the values lie on either side of it ]
Median:
µ
µµ
-
=
?
med
med x med
d
n
P x dx
Def
(b)
Median
Median
(c)
Median
(a)
[ ]
[ ] [ ]
[ ] [ ] [ ]
aa
aa
aa aa
aa aa
--
-- --
+= +
+ +
? += +
??
?? ??
L ,  E ( ) ( , )
=   ( , ) y ( , ) =E E

E E E

et X Y x y f x y dxdy
x f x y dxdy f x y dxdy X Y
XY X Y
[ ]
aa
aa
a
a
--
-
=
??
?
(X, Y) an independent RVs with a joint pdf of f(x,y)
E , ( , )
For independent variable f(x,y) = g(x)  ( ) ( x and y are indpendent)
=   (
X Y xy f x y dxdy
x h y because
xy g x
[ ] [ ]
[ ] [ ] [ ]
a aa
a aa - --
•+
? += ×
? ??
) ( ) = ( ) y h( ) = E E

E E   E  (If x and Y are independent)

h y dxdy x g x dx y dy X Y
XY X Y
Module 7
Measures of Central Tendency                            Contd…
Range : (Max - min) value
a
a
µ
µ
s
-
-
?
2
2
2
Most important measure of dispersion
Variance:
the mean
=  ( ) ( )
=
x f x dx
= E(X)
= Expected value of x
Module 7
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