Agriculture Reference
In-Depth Information
No of insect
4.5-5.5
5.5-6.5
6.5-7.5
7.5-8.5
9.5-10.5
10.5-11.5
11.5-12.5
Frequency
14
18
23
16
11
19
4
Solution. From the given information, we make
the following table:
7. Geometric mean can't be calculated if any of
the observation is zero.
8. GM is useful in the construction of index
numbers.
9. As GM gives greater weights to smaller items,
it is useful in economic and socioeconomic
data.
X
i
Frequency(
f
i
)
Log (
X
i
)
f
i
. Log (
x
i
)
5
14
0.69897
9.78558
6
18
0.77815
14.0067
7
23
0.84509
19.4370
8
16
0.90309
14.4494
9
11
0.95424
10.4966
Harmonic Mean
The harmonic mean of a set of “
10
19
1
19.0000
11
4
1.04139
4.16556
” observations
is the reciprocal of the arithmetic mean of the
reciprocals of the observations.
Let
n
f
1
X
n
i¼
1
1
P
n
91
340
105
:
number of
observations. So the arithmetic mean of
x
1
,
x
2
,
...
,
x
n
be
the
n
Thus
;
log
ðGÞ¼
log
ðx
i
Þ¼
the
1
f
i
P
n
i¼
1
x
i
reciprocals of the observations is
. Hence,
¼
0
:
869
i¼
1
n
n
P
n
the harmonic mean, HM
¼
.
G ¼
Alog
ð
0
:
869
Þ¼
7
:
396
:
1
x
i
i¼
1
Properties
1. It is rigidly defined.
2. It is not so easy to calculate or understand the
physical significance.
3. It takes care of all observations.
4. Mathematical treatments are not as easy as in
the case of arithmetic mean.
If
For grouped data:
Let
”
number of classes with the respective frequencies
of
x
1
,
x
2
,
...
,
x
n
be the mid values of the “
n
f
1,
f
2,
f
3,
...
,
f
n
; then, the harmonic mean is
P
i¼
1
f
i
given by
P
i¼
1
f
i
=x
i
:
G
k
are the geometric means of
the samples having
G
1
,
G
2
,
...
,
Example 8.7.
Find the harmonic mean of the
following observations: 15,20,25,30,35,40,45,
and 50.
n
k
numbers of
the observations, then the combined geomet-
ric mean is given by
n
1
,
n
2
,
...
,
Solution. Here, the number of observations is 8.
So their harmonic mean is given by
1
=
P
k
=
P
k
Y
k
i¼
1
G
i¼
1
n
i
1
i¼
1
n
i
G ¼ðG
n
1
1
:G
n
2
...G
n
k
Þ
¼
n
P
n
8
P
i
HM
¼
¼
1
x
i
8
1
x
i
X
k
i¼
1
n
i
log
1
P
or log
G ¼
½G
i
:
i¼
1
i¼
1
1
i¼
1
n
i
8
¼
1
15
þ
1
20
þ
1
25
þ
1
30
þ
1
35
þ
1
40
þ
1
45
þ
1
50
5. If all observations are equal to a constant, say
A
8
.
6. Geometric mean is not influenced very much
by the inclusions of a few large or small
observations unlike the arithmetic mean.
, then the geometric mean is also equal to
A
¼
2857
¼
28
:
00
:
0
:
Example 8.8.
Find the harmonic mean from the
following frequency distribution.
