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Propagating Uncertainties
Chemistry BC3368y
Every experimental quantity
x
has associated with it an uncertainty
If the quantity is reported as
x ±
Δ
x
, then
Δ
x is the
absolute uncertainty
Absolute uncertainties have the same units as x.
Absolute uncertainties are generally written with one nonzero digit,
with x rounded to the same decimal place.
Δ
x/x is called the
relative uncertainty
.
It is dimensionless.
Where does
Δ
x come from?
Sometimes simple estimation, sometimes it is the standard
deviation from a series of replicate determinations. This is often denoted
σ
x
.
When a quantity z is the result of a calculation involving one or more quantities that have
uncertainties, then the uncertainty of the result must be calculated by suitable propagation.
In labs in General Chemistry (BC2001x) and in Quantitative Techniques (BC3340y) you were
often told to use a
worstcase scenario
for error (or uncertainty) propagation: you add them.
The important rules you learned then were the following:
When adding or subtracting, the absolute uncertainty of the result is the sum of
the absolute uncertainties of the terms: If z = x + y, then
Δ
z =
Δ
x +
Δ
y
When multiplying or dividing, the relative uncertainty of the result is the sum of
the relative uncertainties of the terms: If z = x y, then (
Δ
z/z) = (
Δ
x/x) + (
Δ
y/y)
A corollary of the latter is that if you multiply by an exact constant, the relative uncertainty is
unchanged: if
z = ax
, where a is a constant, then
(
Δ
z/z) = (
Δ
x/x)
because
Δ
a = 0.
It follows that
Δ
z = z (
Δ
x/x) = (a x)
Δ
x/x = a
Δ
x.
In other words, the absolute uncertainty is multiplied by the same constant:
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 Spring '10
 Carter
 mechanics

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