Math 211
Introduction to
Statistics
Chapter III
The Measures of Central Tendency
Index (subscript) Notation:
Let the symbol
i
X
(read ‘
X
sub
i
)
denote any of the
N
values
1
2
,
,...,
N
X X
X
assumed by a variable
X
. The letter
i
in
i
X
(
29
1,2,.
..,
i
N
=
is called an index
or subscript. The letters
, , , or
j k p q
s
can also be used.
Summation Notation:
1
2
1
....
N
i
N
i
X
X
X
X
=
=
+
+
+
å
Example :
1 1
2 2
1
....
N
iı
N
N
i
X Y
X Y
X Y
X Y
=
=
+
+
+
å
(
)
1
2
1
2
1
1
....
....
N
N
i
N
N
i
i
i
aX
aX
aX
aX
a X
X
X
a
X
=
=
=
+
+
+
=
+
+
+
=
å
å
,
a
Î ¡
Averages (Measures of Central Tendency)
The average of a set of numbers is the value which best represents it. There are three different
types of averages. Each has advantages and disadvantages depending on the data and intended
purpose.
Mean
This is also known as the arithmetic mean. It is found by dividing the sum of the set of
numbers with the actual number of values and defined as
1
2
1
...
N
i
N
i
X
X
X
X
X
X
N
N
N
=
+
+
+
=
=
=
∑
∑
Example :
Find the mean of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.
Sum of values: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
Number of values = 10
Mean of values=
X
= 55 / 10 = 5.5
Note:
If the numbers
1
2
,
,...,
k
X X
X
occur
1
2
,
,...,
k
f f
f
times respectively, (occur with
frequencies
1
2
,
,...,
k
f f
f
), the arithmetic mean is,
1
k
i
i
i
f X
X
N
=
=
∑
where
1
k
i
i
N
X
=
=
∑
is the total frequency.
1