Error Propagation
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3 Types of Measurements
For the purposes of error analysis, there are 3
categories of measurements:
1. Direct, non-repeatable
2. Direct, repeatable
3. Indirect (calculated)
Direct, Non-Repeatable Measurements
• Uncertainty in one such measurement is
determined by the precision and accuracy of
the measuring instrument + other factors
affecting the experimenter’s ability to make
the measurement.
▫ Factors like parallax (difference between what
each of your 2 eyes sees)
• Experimenter must use their best judgment to
estimate and report the uncertainty as:
▫ Measurement = value ± uncertainty [units]
Direct, Repeatable Measurements
• Any direct, repeatable measurement should be
taken many times in succession to calculate
the standard deviation:
• The standard deviation is always preferred to
the estimated uncertainty of a single
measurement
▫ With it you calculate the spread of your data
instead of guessing at it
Indirect Measurements
• Let’s say f is a function of directly measureable
variables x and y, each carrying their own
uncertainties, i.e.
▫ f = f(x, y) with x = x ± σx & y = y ± σy
• Uncertainty in f can only be determined by
propagating σx and σy
▫ Procedure is roughly the same whether the
component uncertainties are standard
deviations or estimates for non-repeatable
measurements
General Propagation Formula
• Uncertainties typically add as the root-sum-ofsquares (“RSS”; can be derived with
calculus)
▫ The following assumes x and y are
independent of each other:
Special Cases:
•Notice that
the propagated
errors for both
f = xy and f =
x/y are the
same
•Adding or
multiplying by
a constant does
not affect the
error
• Other forms
of the σf can
be found online
Secret Time-Saver
• If one uncertainty term is more than ~3 times
greater than any of the others, the other
terms will usually drop out when you round
to the correct significant figures
▫ Thus, the combined uncertainty will be
approximately equal to the largest
uncertainty
If you absolutely can’t calculate σf
• …Then you can use the upper-lower-bound
method:
▫ Use the minima and maxima of each variable
to calculate the minimum and maximum
values of f(x, y, …)
• This method overestimates the error in f, thus
is primarily useful when f is incompletely
defined or when the variables x and y have
an unknown correlation