Agriculture Reference
In-Depth Information
8.2.1.1 Mean
Means are of three different types:
P
k
i¼
1
n
i
x
i
P
k
i¼
1
n
i
arithmetic
x ¼
:
mean, geometric mean, and harmonic
mean.
Arithmetic Mean
The arithmetic mean of a set of observations is
their sum divided by a number of observations
and is denoted by
x
or
μ
, that is,
Example 8.3.
The following table gives the
number of students and their average height
(cm.) of the four different classes of under-
graduate courses along with their respective
means. Find the overall average age of the
undergraduate students.
n
i¼
1
x
i
n
x ¼ μ ¼
x
1
; x
2
; ...; x
n
where
are the values of the first,
1st
year
2nd
year
3rd
year
4th
year
second, third,
th observation.
The arithmetic mean for grouped/classified
...
,
n
Class
No. of students
135
132
140
135
P
i¼
1
f
i
x
i
;
P
i¼
1
f
i
¼ N
,
Average height in cm.
166
175
188
190
P
n
i¼
data is given by
x ¼
1
f
where x
i
's
and
f
i
'
s are the mid values and the
The overall average age of the undergraduate
students are given by
frequencies of the
i
th (
i ¼
1, 2, 3,
...
,
n
) class.
P
k
i¼
1
n
i
x
i
P
k
i¼
1
n
i
Properties of Arithmetic Mean
1. It is readily defined.
2. It is easy to calculate.
3. It is easy to understand.
4. It is based on all observations.
5. It is readily acceptable for mathematical
treatments.
6. The arithmetic mean of the “
x ¼
;
here
k ¼
4
;
so
x ¼½
135
166
þ
132
175
þ
140
188
þ
135
190
=
542
¼
179
:
85 cm
:
n
” number of
9. Arithmetic mean cannot be worked out
simply by an inspection of data.
10. Arithmetic mean is highly affected by the
missing observations or a few large or small
observations.
11. Arithmetic mean sometimes seems to be
misleading, particularly in the presence of
an outlier.
12. AM cannot be calculated if any of the obser-
vation is missing.
13. AM cannot be used for open-ended classes
because in that case, it is difficult to get the
mid value of the class.
) is also the constant.
7. Arithmetic mean depends on the transforma-
tion of data. If
constants (say
x ¼ A
be the transfor-
mation of
X
data to
Y
; where
Y ¼ a
+
bX
a
and
b
are
constants and
x
is the arithmetic mean of the
, then
Y ¼ a þ bx:
Let us suppose that variable
variable
X
has an arithme-
tic mean of 57 and that this variable is related
with another variable
X
Y
as
Y ¼
15 + 2.5
X
.
The arithmetic mean of
Y
then would be
157.5
8. The overall mean of “
(15 + 2.5
57)
¼
k
” number of arithmetic
means from “
k
” number of samples with
n
1
,
n
2
,
n
3
,
...
n
k
and the arithme-
observations
,
Geometric Mean
The geometric mean of a set of “
n
” observations
is defined as the
tic means
x
1
; x
2
; x
3
; ...; x
k
are the weighted
average of the arithmetic means. Thus, for
n
th root of the product of all
observations.
Let
...
k
Samples
1
2
3
4
5
x
1
,
x
2
,
...
,
x
n
be the “
n
” number of
No. of observations
n
n
n
n
n
...
n
k
observations of variable “
X
”;
the geometric
1
2
3
4
5
AM
x
1
x
2
x
3
x
4
x
5
...
x
n
mean then is given by
