Agriculture Reference
In-Depth Information
8. The combined variance of “
k
” number of
N
i¼
1
ðx
i
AÞ
r
for raw data and for grouped data
1
N
μ
0
r
ðAÞ¼
X
samples with
n
1
,
n
2
,
n
3
,
...
,
n
k
and
1,
X
X
,
Xk
2,
3,
as number of observations
and means,
respectively,
is given by
h
i
,
P
i¼
1
n
i
σ
i
þ
P
i¼
1
n
i
d
X
n
i¼
1
f
i
ðx
i
AÞ
r
1
2
P
k
i¼
1
2
2
i
σ
¼
μ
0
r
ðAÞ¼
n
i¼
1
f
i
1
n
i
2
i
where
σ
is the variance of
the
i
th
sample with “
n
i
” observations, and by
where variate
X
takes the values
x
i
(
i ¼
1,2,
d
i
¼ x
i
X
X
, where
is the combined
3,4,
...
,
N
) for raw data and the mid value
x
i
mean of all the samples.
(
i ¼
1,2,3,4,
...
,
n
) with respective frequency
of
f
i
(
i ¼
1,2,3,
...
,
n
) for
n
classes/groups and
Example 8.17.
The following data gives the gen-
eral features of plant height (ft) of maize plants
for two different samples. Find the composite
variance of the maize plants.
P
i¼
1
f
i
¼ N:
If we take
A ¼
0, then we get the moment
about the origin
X
n
1
1
f
i
x
i
υ
r
¼
and putting
A ¼ x;
and
Characteristics
Sample 1
Sample 2
n
i¼
1
f
i
i¼
Sample size
65
35
Mean height (ft)
5.34
5.49
X
n
i¼
1
f
i
x
i
x
1
Sample variance (ft
2
)
3.89
2.25
Þ
r
;
m
r
¼
ð
n
1
f
i
i¼
Solution. Combined mean height is given by
X ¼ n
1
X
1
þ n
2
X
2
the
r
th central moment
:
½
= n
1
þ n
2
½
¼
5
:
4
,
and com-
2
1
bined
variance
is
given
by
σ
¼
P
2
i¼
1
n
i
It can be noted that
h
i
P
2
i¼
1
n
i
σ
i
þ
P
2
μ
0
0
ðAÞ¼υ
0
¼ m
0
¼
2
2
1 and
So one can
conclude that the average height and the com-
bined variance of the samples are 6.4 ft and
2.21 ft
2
, respectively.
i¼
1
n
i
d
i
¼
2
:
21
:
2
υ
1
¼ x; m
1
¼
0,
m
2
¼ σ
¼ d
r
m
r
(
If
y ¼
(
xc
)/
d
, then
m
r
(
x
)
y
), where
“
c
” and “
d
” are const
an
ts. Sinc
e
x
i
¼ c
+
dy
i
,
Example 8.18.
If the relation between two
variables is given as
i ¼
1,2,3,4,
...
,
n
, and
x ¼ c þ dy;
we have
Y ¼
25 + 4.5
X
and the
X
n
i¼
1
f
i
ðc þ dy
i
c d YÞ
r
1
2
σ
X
¼
4
:
2, then find the standard deviation of
Y
.
m
r
ðxÞ¼
n
i¼
1
f
i
2
2
2
X
Solution. We know that
σ
Y
¼ b
σ
, here
2
b ¼
σ
2
Y
¼
ðÞ
:
:
¼
:
:
X
n
4.5, so
4
5
4
2
85
05
1
σ
Y
¼þ
p
85
1
f
i
d
r
ðy
i
YÞ
r
¼ d
r
m
r
ðyÞ
¼
:
Thus, from the discussion of the above
measures of dispersion, one can find that the
standard deviation/variance follows almost all
the qualities of a good measure. As a result of
which, this is being extensively used as measure
of dispersion.
(e) Moments
The
Standard deviation
ð
:
05
Þ¼
9
:
22
n
i¼
1
f
i
i¼
8.2.2.2 Relative Measures of Dispersion
Relative measures of dispersions are mainly
coefficients based on the absolute measures,
also known as coefficients of dispersion, are
unit-free measures, and are mostly ratios or
percentages.
1.
r
th raw moment about an arbitrary point
“
th power of the
deviation of the observations from the point
“
A
” is the mean of the
r
Coefficient of dispersion based on range
:
It is defined as
X
max
X
min
X
max
þX
min
;
X
max
and
X
min
are
A
” and is denoted as