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M. Palmer
1
Propagation of Uncertainty through Mathematical Operations
Since the quantity of interest in an experiment is rarely obtained by measuring that quantity
directly, we must understand how error propagates when mathematical operations are performed
on measured quantities.
Suppose we have a simple experiment where we want to measure
velocity,
V
, by measuring distance,
d
, and time,
t.
We take measurements and come up with
measured quantities
d
±
δ
d
and
t
±
δ
t
.
We can easily estimate
V
by dividing
d
by
t
, but we also
need to know how to find
δ
V
.
Below we investigate how error propagates when mathematical
operations are performed on two quantities
x
and
y
that comprise the desired quantity
q
.
Addition and Subtraction
If we are trying to find the uncertainty,
δ
q
, associated with
q
=
x
+
y
, we can look at what the
highest and lowest probable values would be.
The highest value would be obtained by adding
the best estimates for
x
and
y
to the total uncertainty for both values.
Similarly, the lowest
probable value would be obtained by adding the best estimates for
x
and
y
and subtracting both
associated uncertainties:
(highest probable value of
q
) =
x
best
+
y
best
+ (
δ
x
+
δ
y
)
(2)
(lowest probable value of
q
)
=
x
best
+
y
best
_ (
δ
x
+
δ
y
)
(3)
Since
q
best
=
x
best
+
y
best
, it is easy to see
δ
q
is equal to
δ
x
+
δ
y
and
q
can be expressed as:
q
best
=
x
best
+
y
best
± (
δ
x
+
δ
y
)
(4)
Similarly, for subtraction, we can write that:
q
best
=
x
best
_
y
best
± (
δ
x
+
δ
y
)
(5)
A similar analysis as shown above can be applied to any number of quantities that are added or
subtracted, and we can state a general rule:
For
q
=
x
+ … +
z
– (
u
+ …. +
w
),
δ
q
=
δ
x
+ …+
δ
z
– (
δ
u
+ … +
δ
w
)
Or more simply put, when adding or subtracting quantities, their uncertainties add.
Uncertainty of a Product
We can perform a similar analysis as above for multiplication, but first we must define fractional
uncertainty of a quantity:
Rule 1: Uncertainty in Sums and Differences
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View Full DocumentM. Palmer
2
(fractional uncertainty in x) =
best
x
x
δ
.
(6)
The fractional uncertainty (or, as it is also known, percentage uncertainty) is a normalized,
dimensionless way of presenting uncertainty, which is necessary when multiplying or dividing.
A measurement and its fractional uncertainty can be expressed as:
(value of x) =
+
best
best
x
x
x
1
.
(7)
For simplicity, hereafter the subscript ‘best’ will be omitted in the denominator of the fractional
uncertainty, but it is assumed.
For
q
=
xy
, we have measured values for
x
and
y
of the form:
(measured x) =
+
x
x
x
best
1
(8)
(measured y) =
+
y
y
y
best
1
(9)
Once again we look at the largest and smallest probable values of
q
:
(largest probable value of
q
) =
+
+
y
y
x
x
y
x
best
best
1
1
(10)
(smallest probable value of
q
)
=
−
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 Spring '08
 ERSKINE/TSOI

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