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Error Propagation (Propagation of Uncertainty) - Statistics How To
Statistics How To
Error Propagation (Propagation of Uncertainty)
Measurement Error >
Error propagation (or propagation of uncertainty) is what happens to measurement errors when you use those uncertain
measurements to calculate something else. For example, you might use velocity to calculate kinetic energy, or you might use
length to calculate area. When you use uncertain measurements to calculate something else, they propagate (grow much
more quickly than the sum of the individual errors). To take this propagation into account, use one of the following formulas in
your experiments.
These formulas assume your errors are random and not correlated (e.g. if you have systematic errors, you can’t use them).
Error Propagation Contents:
1. Addition or Subtraction Formula
2. Multiplication or Division formula
3. Measured Quantity Times Exact Number formula
4. General formula
5. Power formula
6. Error Propagation in Calculus
1. Addition or Subtraction
Additive formula for error propagation.
Where:
a,b,c are positive measurements
x,y,z are negative measurements
δ is the error associated with each measurement (the absolute error). δa is the uncertainty associated with measurement
a, δb is the uncertainty associated with measurement b, and so on.
Example of Worked Formula
Let’s say you measured your height (a) as 2.00 ± 0.03 m. Your waistband (b) is 0.88 ± 0.04 m from
the top of your head, which means your pant length P would be p = H – w = 2.00 m – 0.88 m = 1.12
m.
The uncertainty, using the addition formula, is:
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Error Propagation (Propagation of Uncertainty) - Statistics How To
Giving a final measurement of 1.12 m ± 0.05 m.
2. Multiplication or Division formula
When calculating errors, there is no difference between multiplication and division.
3. Power formula
If n is an exact number and Q = xn, then
4. Measured Quantity Times Exact Number formula
If A is exact measurement (e.g. A = 9 or A = π) and Q = Ax, then:
δQ = |A| δx
5. General formula for Error Propagation
You might wonder why you can’t just add (or multiply, or divide) the errors and be done with it. Why do we have to use
formulas? Very basically, one small measurement error on an independent variable, when applied to a function (say, a formula
for area, kinetic energy, or velocity) is going to result in a much larger error on the dependent variable.
Error Propagation in Calculus
Why the formulas work requires an understanding of calculus, and particularly derivatives; They are derived from the Gaussian
equation for normally-distributed errors. If you have some error in your measurement (x), then the resulting error in the function
output (y) is based on the slope of the line (i.e. the derivative).
The general formula (using derivatives) for error propagation (from which all of the other formulas are derived) is:
Where Q = Q(x) is any function of x.
Error propagation formulas are based on taking partial derivatives of a function with respect to the variable with the uncertainty.
Let’s say you had a function with three variables (x, u, v) and two of those (u, v) have uncertainty. The variance of x can be
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Error Propagation (Propagation of Uncertainty) - Statistics How To
approximated by [1]:
Example question: The volume of gasoline delivered from a pump is the difference between the initial (I) and final (F) readings.
If each reading has an uncertainty of ±0.02mL, what is the error in the volume delivered?
Solution:
V = F – I; σ2(V)
= σ2(I) + σ2(F)
= (0.02mL)2 + (0.02mL)2
= 0.0008mL2
= 0.028mL
The error in the volume delivered is 0.028mL.
This calculation will only work when the partial derivatives are 1. The formula changes slightly when x is a product (x = uv) or
quotient (x = u/v):
Example question #2: A shipping container is 12 x 10 x 8 feet with an uncertainty of 0.1ft. What is the uncertainty in the
volume?
Step 1: Calculate the volume:
V = 12 x 10 x 8 = 960 ft3.
Step 1: Work the formula:
(0.1 ft)2 / (12 ft2) +
(0.1 ft)2 / (10 ft2) +
(0.1 ft)2 / (8 ft2) =
0.0.00145149572
Step 3: Find the volume variance (Step 1 * Step 2):
(0.00145149572)(960 ft3)2 = 1337.698 ft6
Step 4: Take the square root of Step 3 to find the uncertainty in the volume:
√(1337.698) = 36.57 ft3.
References
10. Error Propagation tutorial.doc. Retrieved April 14, 2021 from:
https://foothill.edu/psme/daley/tutorials_files/10.%20Error%20Propagation.pdf
CITE THIS AS:
Stephanie Glen. "Error Propagation (Propagation of Uncertainty)" From StatisticsHowTo.com: Elementary Statistics for
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