Agriculture Reference
In-Depth Information
1 =n
Y
Solution. There are eight numbers of
observations. If we denote the geometric mean
by
1 =n
X g ¼ðx 1 :x 2 ...x n Þ
¼
x i
Taking logarithm of both the sides, we have
G
, then
G ¼
(20
24
28
32
36
48) 1/8
40
1/8[Log(20) +
Log(24) + Log(28) + Log(32) + Log(36) + Log
(40) + Log(44) + Log(48)]
44
¼ >
Log(
G
)
¼
log Y
1
n
log
ðX g Þ¼
x i
1
n :
x 2 þ ... þ
¼
½
log
x 1 þ
log
log
x n
x
Log(
x
)
20
1.3010
X n
1
n
24
1.3802
¼
log
x i ¼ Að
say
Þ
28
1.4472
1
32
1.5051
so,
ðAÞ:
Thus, the logarithm of a geometric mean is the
arithmetic mean of logarithm of the observations.
For grouped/classified data on variable
X g ¼
Antilog
ð
log
x g Þ¼
Antilog
36
1.5563
40
1.6021
44
1.6435
48
1.6812
X
hav-
ing
x n as class mid values with the
respective frequencies of
x 1 ,
x 2 ,
...
,
Thus, log(
G
)
¼
1/8 (12.1166)
¼
1.5146
f 1,
f 2,
f 3, ...
,
f n ,
the
So,
the antilog of
log(
G
)
¼ G ¼
antilog
geometric mean is given by
(1.5146)
32.7039. So the geometric mean of
the given numbers is 32.7039.
¼
1 = P n
1 = P n
Y
1 f i
1 f i
X g ¼ x f 1 :x f 2 ...x f n
x i f i
¼
:
Example 8.5.
Find the geometric mean from the
following frequency distribution.
For grouped frequency data,
x i
is taken as the
mid value of the i th class.
With the help of a log conversion or scientific
calculator, one can easily find out the geometric
mean.
Variable values 15 20 20 25 30
Frequency 3 4 3 4 6
Solution. This is a simple frequency dist-
ribution and we have the geometric mean
1 = P n
Example 8.4.
Find the geometric mean of the
following observations: 20,24,28,32,36,40,44,
and 48.
1 f i . Taking the logarithm
of both sides, we have Log(
X g ¼ x f 1 :x f 2 ...x f n
x g )
!
log Y n
1
P n
1 f i
1
20 3 log
1 x 1 fi
¼
½
ð
15
Þþ
4 log
ð
20
Þþ
3 log
ð
20
Þþ
4 log
ð
25
Þþ
6 log
ð
30
Þ
1
20 3
¼
½
1
:
1760
þ
4
1
:
3010
þ
3
1
:
3010
þ
4
1
:
3979
þ
6
1
:
4771
Þ
1
20 27
¼
½
:
0892
¼
1
:
35446
:
X g ¼
Alog 1
ð
:
35446
Þ ¼
22
:
618
:
Thus, the geometric mean of the above simple
frequency distribution is 22.618.
Example 8.6.
Find the geometric mean from the
following frequency distribution of the stem borer.
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