Download Advanced Uncertainty Propagation and Error Analysis

Brown University
Physics Department
PHYS 0030
Analysis Of Experimental Uncertainties
ADVANCED UNCERTAINTY PROPAGATION AND ERROR
ANALYSIS
This document is meant to supplement “Uncertainty Propagation and Error Analysis” (which can
be found on the Physics 0030 lab wiki) to cover more advanced aspects of uncertainty
propagation and error analysis.
Calculus-Based Uncertainty Propagation
If a student aspires to do rigorous uncertainty propagation, and feels comfortable with calculus,
then the student is encouraged to use a calculus technique that simplifies (and avoids certain
pitfalls in) uncertainty propagation calculations.
Suppose we have taken measurements x1, x2,…, xN with respective uncertainties Δx1, Δx2,…, ΔxN.
We then use these measured values to calculate the quantity z, where z is given as a function f
of the variables x1, x2,…, xN, i.e., z = f(x1, x2,…, xN). Assume that the variables x1, x2,…, xN are
independent. Then the uncertainty in z, denoted Δz, can be calculated from the uncertainties Δx1,
Δx2,…, ΔxN according to:
Where we have taken the partial derivatives of f with respect to each x variable. Note that this
technique is often a quicker way to propagate uncertainty and avoids a common pitfall that may
arise from applying the algebraic formulae: when the same variable xi appears more than once in
some function for the quantity z, then it is common to incorrectly treat the uncertainty Δxi in one
xi term as independent of the uncertainty in the other xi terms when using the algebraic formulae.
This calculus formula ensures that such problems do not arise, and if a student notices a
discrepancy between the two methods, then she should investigate whether this common mistake
occurred. However, making this mistake typically does not dramatically change the resulting Δz,
and thus an otherwise correct application of the algebraic formulae may serve as an acceptable
estimate of the propagated uncertainty Δz.
Normally Distributed Random Variables
We recall the formulae for the average and sample standard deviation of N measured values of
some quantity h:
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Brown University
Physics Department
PHYS 0030
Analysis Of Experimental Uncertainties
These statistics can be applied to any set of data, regardless of how the individual measurements
hi are distributed. For example, if we repeatedly performed the experiment of rolling a 6-sided
die and recording which number it shows, then (unless the die is somehow loaded or biased) we
would expect each of the six possible values to occur roughly the same number of times. This is
an example of a variable that is uniformly distributed. However, if we repeatedly performed the
experiment of rolling two dice and recording their sum, then it turns out that the value 7 will
probably be more common than any other value. In this case, the distribution of the values is not
uniform because 7 is more likely than other possible values like 2 or 12.
One distribution that is quite interesting is called the Normal Distribution. Many variables are
observed to be distributed approximately Normally, including: the sum of a large number of dice
rolls, the height of US adult males and the height of US adult females, SAT scores, and the
position of a ground-state quantum particle in a harmonic potential. The Normal distribution can
be written in terms of its probability distribution function, given by:
In this expression, P(x) is associated with the probability that the random variable x takes on a
given value; in particular, the probability that the random variable x lies in the range bounded by
xi and xf is given by the definite integral of P(x) with respect to x from xi to xf. It turns out that for
random variables x distributed according to the above Normal distribution, the average of
measured values x1, x2,…, xN will be approximately equal to µ and the sample standard deviation
will be approximately equal to σ, where these approximations become increasingly exact as the
number N of measured x values increases.
The following is a plot of the above Normal distribution indicating the probabilities that a given
x measurement lies within various intervals of width σ centered around the mean µ.
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Brown University
Physics Department
PHYS 0030
Analysis Of Experimental Uncertainties
The x-axis shows the possible values of x, and the y-axis is related to the relative frequency of a
particular value. The maximum of the curve is the average. The width of this curve is associated
with the standard deviation σ, which can be estimated from a finite sample using the sample
standard deviation s. A large standard deviation yields a wide curve, while a small standard
deviation yields a narrow curve.
For example, suppose a measurement of x is drawn from a population distributed according to
the above distribution. The probability that the value would fall within 1 standard deviation of µ
is 68%, and there is a 98% probability that it would fall within 2 standard deviations of µ.
Suppose we made N measurements of the quantity x distributed according to the above
distribution, and then we calculated their average x̅ . Now suppose we repeat this process M
times, drawing N new x values each time and computing their average x̅ . It is possible to show
that these averages will also be approximately distributed Normally, and their distribution will be
more exactly Normal as we increase M. In the limit of large M, it turns out that the Normal
distribution of x̅ will have the same average µ, but the standard deviation of this new distribution
will be σ/sqrt(N). Thus if we take any one of the computed x̅ values, there is a 68% probability
that it will lie within σ/sqrt(N) of µ, a 98% probability that it will lie within 2σ/sqrt(N) of µ, etc.
This shows that as one increases the number of measurements N, their average will more likely
be closer to the population’s true average.
In practice, the population’s true mean µ and true standard deviation σ are typically not known
and must be estimated using a finite sample. Based on the considerations in the last paragraph,
the probability that a sample average x̅ is close to the population’s true average µ is related to the
quantity σ/sqrt(N), but with a finite sample, we do not know the sample’s true standard deviation
σ. So to estimate σ/sqrt(N) from our finite sample, we use the sample standard deviation s as an
estimate for σ. This yields a statistic called the Standard Error SEx̅ which is given by SEx̅ =
s/sqrt(N). Thus, for a finite sample of N measurements of a quantity x, the standard error SEx̅
gives us an approximation for the distribution of possible sample averages, and using this
information, we can use the properties of Normal distributions to infer how probable it is for the
true average µ to be close to an experimentally determined sample average x̅ .
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