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Comparing with Percentages
Using relative differences (i.e. percentages) rather than absolute differences is important
because the significance of an absolute difference depends on the sizes of the quantities
involved. The percentage in question is always a percentage of something.
Experimental Percent Error – In this case you’re evaluating a new measured value as
compared to a previously known theoretical or accepted value. The “Error” is the
absolute value of the difference between the two values.
% Error = (
|Error|
Accepted
|Measured − Accepted|
Accepted
Percent Difference – In this case you have two values that are on equal footing. Then
there is no good way of choosing which belongs in the denominator. As a compromise,
place the average in the denominator.
% Difference = (
|Difference|
Average
|Val2 − Val1|
(Val1 + Val2)/ 2
Percent Change – In this case, some value is changing, and there is a new value and an
old value. Here, it may be interesting to note whether the value increased or decreased,
so an absolute value isn’t usually taken. Assuming positive quantities, a positive change
is an increase and a negative change is a decrease.
% Change = (
New − Old
Old
Reference:
http://webassign.net/question_assets/ncsucalcphysmechl3/percent_error/manual.html
Uncertainty
A measurement uncertainty is a possible error in a measurement. It’s expected that the actual
error in the measurement is less than the uncertainty. Uncertainty can be expressed as an
absolute quantity (in the same units as the measurement) or as a relative quantity (i.e. a
percentage). Uncertainty is always positive, even though we don’t know which direction the
error is in.
There are a few examples below.
Propagation of Uncertainty
How do uncertainties combine to produce results with uncertainty? Generally, when adding or
subtracting, add the uncertainties. When multiplying or dividing, add the relative uncertainties.
Quick Examples of Uncertainty
Quantities can be expressed with absolute or relative uncertainty.
x = 50 ± 1 m = 50 m ± 2%
y = 20 ± 1 m = 20 s ± 5%
t = 25 ± .5 s = 25 s ± 2%
When adding or subtracting, add the uncertainties.
x – y = 30 ± 2 m
When multiplying or dividing, add the relative uncertainties.
xy = (50 m ± 2%) (20 m ± 5%) = (1000 m
2
± 7%) = (1000 ± 70 m
2
)
x / t = (50 m ± 2%) / (25 s ± 2%) = (2 m/s ± 4%) = (2 ± .08 m/s)
With powers, scale the relative uncertainty by the power. (Square root is the one-half power.)
t
3
= (25 s ± 2%)
3
= (15625 s
3
± 6%) = (15625 ± 937 s
3
) ≈ (15600 ± 900 s
3
)
√( xy ) = √(1000 m
2
± 7%) = (31.62 m ± 3.5%) = (31.62 ± 1.1 m) ≈ (31.6 ± 1 m)
Remember that when you add or scale a relative uncertainty, the result is still a relative uncertainty.
Description Operations
Numerical Uncertainty
(Same units as value)
Quantity = Val ± Unc
Relative Uncertainty
(Numerically dimensionless, but written as %, ppm, or
ppb)
Quantity = Val ± %Unc
or
Addition and Subtraction:
Add Numerical Unc
Value = x 1
Unc = Unc 1 + Unc 2 + ...
Multiplication and Division:
Add %Unc (or RelUnc)
Value = x 1 x 2
%Unc = %Unc 1
Powers and Roots:
Scale %Unc (or RelUnc)
Value = x 1
Power
/ x 2
Power
%Unc = %Unc 1 ×Power 1 + %Unc 2 × Power 2 +
