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Error Propagation (Propagation of Uncertainty)

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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

1. Addition or Subtraction

error propagation

Additive formula for error propagation.


  • 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:

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

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.
The derivation of the individual formulas is beyond the scope of this article. However, you don’t need to understand the calculus to use the formulas.

Stephanie Glen. "Error Propagation (Propagation of Uncertainty)" From Elementary Statistics for the rest of us!

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