measurement equals the same value of
σ
/
μ
·
100%. One step further is to say that
the accuracy of the calibrated current meter is
σ
/
μ
·
100%. Usually this is only
valid within a certain range of measurements as a different accuracy might apply
to extreme low or high current.
2.3
E R RO R P RO PAG AT I O N F RO M M AT H E M AT I C A L R E L AT I O N S
Often a required parameter is derived from other measurable parameters. For ex
ample, average flow in a river (under stationary and uniform conditions) can be
derived from measuring stage, slope and roughness. Then the question arises how
an error in the measured parameters translates into the error of the wanted parame
ter. This would not only determine the total error but also, which of the parameters
is most critical to the total error and hence should be measured with more care.
Relative simple equations can be derived at, giving the variance
σ
2
of the wanted
parameter expressed in terms of well known statistical terms (variance and co
variance) of the measured parameters. The covariance determines whether or not
measured parameters are statistical dependent.
Suppose that, in order to find a value for the function q(
x
,
y
), we measure the two
quantities
x
and
y
. The standard error of
σ
2
becomes:
σ
2
q
=
∂
q
∂
x
2
σ
2
x
+
∂
q
∂
y
2
σ
2
y
+
2
∂
q
∂
x
∂
q
∂
y
σ
xy
(2.1)
This gives the standard deviation
σ
q
, whether or not the measurement of
x
and
y
are independent, and whether or not they are normally distributed. If the measure
ments of
x
and
y
are independent the covariance
σ
xy
will approach zero. With
σ
xy
zero, the equation reduces to:
σ
2
q
=
∂
q
∂
x
2
σ
2
x
+
∂
q
∂
y
2
σ
2
y
(2.2)
When the covariance
σ
xy
is not zero we say that the errors in
x
and
y
are corre
lated. In this situation the uncertainty
σ
q
in q(
x
,
y
) is not the same as we would get
from the formula for independent errors in
x
and
y
.
2.3
E R RO R P RO PAG AT I O N F RO M M AT H E M AT I C A L R E L AT I O N S
7
Equation 2.1 and 2.2 are the general rules for propagation of errors derived from
mathematical relations. For a number of functions the relation for propagation of
errors has been worked out below:
2.3.0.1
Example: Propagation errors 1
q
(
x
) =
ax
+
b
(2.3)
σ
2
q
=
a
2
·
σ
2
x
(2.4)
2.3.0.2
Example: Propagation errors 2
q
(
x
,
y
) =
a
1
x
+
a
2
y
(2.5)
For an independent relation between
x
and
y
:
σ
2
q
=
a
2
1
·
σ
2
x
+
a
2
2
·
σ
2
y
(2.6)
For a dependent relation between
x
and
y
:
σ
2
q
=
a
2
1
·
σ
2
x
+
a
2
2
·
σ
2
y
+
2
a
1
a
2
σ
xy
(2.7)
The maximum of the covariance
σ
xy
is obtained with maximum correlation
ρ
=
1. In that case
σ
xy
=
σ
x
·
σ
y
. This will be further explained in section 2.4 under
regression correlation analysis.
2.3.0.3
Example: Propagation errors 3
q
(
x
,
y
) =
a
·
x
b
·
y
c
(2.8)
For an independent relation between
x
and
y
:
σ
2
q
=
abx
b

1
y
c
2
σ
2
x
+
acx
b
y
c

1
2
σ
2
y
(2.9)
Through defining the relative errors as follows,
σ
2
q
¯
q
2
=
r
2
q
σ
2
x
¯
x
2
=
r
2
x
σ
2
y
¯
y
2
=
r
2
y
(2.10)
this becomes:
r
2
q
=
b
2
r
2
x
+
c
2
r
2
y
(2.11)
8
T H E O RY O F E R RO R S
2.3.0.4
Example: Slope area method
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 Linear Regression, Regression Analysis, The Land, F E R R O R