Geoscience Reference
In-Depth Information
∑
x
2
= Corrected sum of squares for
X
:
−
(
)
2
∑
X
n
∑
2
X
∑
y
2
= Corrected sum of squares for
Y
:
−
(
)
2
∑
Y
n
∑
2
Y
For the values used to illustrate the covariance we have:
()()
( (
86
395
12
)
∑
()()()()
xy
=
420940
+ ++ −
…
11 40
=
129 1667
.
2
()
86
12
∑
2
2
2
2
y
=+++ −
49
…
11
=
57 6667
.
2
+++− =
(
395
12
)
∑
2
2
2
2
x
=
20
40
…
40
922 9167
.
So:
129 1667
57 6667
.
129 1667
230 6
.
. 980
r
=
=
= .
056
(.
)(
922 9167
.
)
7.10 VARIANCE OF A LINEAR FUNCTION
Routinely we combine variables or population estim
at
es in a linear function. For example, if the
m
ea
n timber volume per acre has been estimated as
X
, then the total volume on
M
acres with be
MX
; the estimate of total volume is a linea
r
function of the estimated mean volume. If the estim
at
e
of cubic volume per acre in sawtimber is
X
1
and that
of
pu
lp
wood above the sawtimber top is
X
2
,
then the estimat
e
of total cubic foot volume per acre is
X
1
+
X
2
. If on a
gi
ven tract the mean volume
per half-acre is
X
1
for spruce and the mean volume per quarter-
ac
re i
s
X
2
for yellow birch, then the
estimated total volume per acre of spruce and birch would be 2
X
+ 4
X
2
. In general terms, a linear
function of three variables (say
X
1
,
X
2
, and
X
3
) can be written as
LaXaXaX
=++
11
22
33
where
a
1
,
a
2
, and
a
3
are constants.
If the variances are
s
2
,
s
2
, and
s
2
(for
X
1
,
X
2
and
X
3
, respectively) and the covariances are
s
1,2
,
s
1,3
,
and
s
2,3
, then the variance of
L
is given by
(
)
2
2
2
2
2
2
2
sasasas
= +++ +
2
a as
aas
+
aas
L
1
1
2
2
3
3
1212
,
1313
,
232,
The standard deviation (or standard error) of
L
is simply the square root of this. The extension of the
rule to cover any number of variables should be fairly obvious.
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