Agriculture Reference
In-Depth Information
X
X
n
i¼
1
f
i
ðX
i
XÞ
variable from the arithmetic mean
of the
1
P
n
2
2
σ
X
¼
n
i¼
1
X
i
X
1
n
variable and is denoted as MD
¼
,
1
f
i
i¼
P
i¼
1
f
i
X
i
X
P
i¼
1
f
i
1
j
j
for grouped data.
X
n
i¼
1
f
i
X
i
1
P
n
i¼
1
f
i
2
X
2
¼
Similarly, the mean deviation from median is
P
n
1
P
i¼
1
f
i
1
n
denoted as MD
Me
¼
1
X
i
Me,
for grouped
=
classified data
:
i¼
P
i¼
1
f
i
X
i
for grouped data, and
mean deviation from mode is MD
Mo
¼
1
n
j
Me
j
Properties of Variance:
1. Variance is rigidly and clearly defined.
2. Variance has the range between 0 to
X
n
i¼
1
X
i
X
n
i¼
1
f
i
X
i
1
P
i¼
1
f
i
Mo,
j
Mo
j
;
when all the observations are equal (i.e.,
the variable remains no longer a variable),
then variance/standard deviation is 0.
3. Variance is based on all observations.
4. Variance cannot be worked out for fre-
quency distribution with open-ended
classes.
5. Variance is amenable to mathematical
treatments.
6. Variance or standard deviation is least
affected by the sampling fluctuations.
7. Variance does not depend on the change
of origin but depends on the change of
scale. That means, two variables “
1
for grouped data
For a grouped frequency distribution,
x
i
is
th class.
(c)
Quartile Deviation
: Quartile deviation is
defined as half of the difference between the
3rd and 1st quartile values QD
taken as the mid value of the
i
¼
Q
3
Q
2
; that
is why, it is also called as semi-inter-quartile
range.
(d)
Standard Deviation
: Standard deviation is
defined as the positive square root of the
arithmetic mean of the square of the
deviations of the observations from arith-
metic mean
and is written as
σ
X
¼
X
” and
s
1
n
n
“
Y
” are related in the form of
Y ¼ a
+
bX
,
1
Xi X
2
þ
ð
Þ
for raw data and
σ
X
¼
where
are two constants known as
the change of origin and scale, respec-
tively; if the variance of “
a
and
b
i¼
s
1
P
n
i¼
1
fiðX
i
XÞ
2
P
i¼
1
fi
þ
2
x
for grouped data,
X
”be
σ
then
2
2
x
the variance of “
Y
” would be
b
σ
where
) are the mid values
of the respective classes with the frequency
of
x
i
(
i ¼
1, 2, 3,
...
,
n
X
n
i¼
1
f
i
Y
i
Y
1
P
n
2
2
σ
y
¼
ð
Þ
) and
X
f
i
(
i ¼
1, 2, 3,
...
,
n
is the mean of
1
f
i
the variable
. The squared quantity of the
standard deviation is known as the
X
i¼
variance
X
n
i¼
1
f
i
a þ bX
i
a b X
1
P
n
i¼
1
f
i
2
¼
ð
Þ
of the variable.
Thus,
is the mean squared deviation
from the mean for a given set of data and is
written as
variance
X
n
1
P
n
i¼
1
f
i
2
2
f
i
X
i
X
2
2
X
:
¼
1
b
ð
Þ
¼ b
σ
i¼
X
n
1
n
1
ðXi XÞ
2
2
X
¼
σ
Thus, the variance of
depends only on
thechangeofscale,notonthechangeof
origin. Similarly,
Y
i¼
X
n
i¼
1
Xi
1
n
2
σ
y
¼
s
:
d
ðyÞ¼ jjσ
X
¼
X
2
¼
for raw data and
jj
s
:
d
:ðxÞ:
