Everyday error propagation problems come down to one relationship between δx, δy, x and y. This calculator turns those numbers into δz and z, with the working shown so you can check the result rather than simply trust it.
What Is the Error Propagation Calculator?
The error propagation calculator turns a set of measurements into δz and z, with the uncertainty made explicit rather than left implied.
Statistics is easy to quote badly. A figure without its spread tells the reader far less than they assume it does, and this calculator puts both together.
How the Error Propagation Is Calculated
The calculation uses the standard relationship for this kind of estimate, applied to the figures you supply. The intermediate steps are exposed so you can see how much of the answer comes from each one.
Sample size matters more than most people expect. Doubling the number of observations improves the precision by a factor of about one and a half, not two.
What the Result Does and Does Not Tell You
A result describes the data you gave it. It does not tell you whether the sample was representative, whether the method was appropriate, or whether there is a confound in the way the figures were collected.
Report the result together with its uncertainty. A number on its own is much easier to misread than a number with a range attached.
A Worked Example
Working through an example, with δx set to 10, δy set to 5, x set to 2, the calculator returns z of 102, and δz of 11.18034, and z (z sub) of -98. Substitute your own figures and the same steps produce the same kind of answer.
Frequently Asked Questions
What do I need to work out δz?
Enter δx, δy, x and y. The calculator then works through the relationship and reports δz and z.
What does the error propagation actually calculate?
It takes δx, δy, x and y and converts them into δz and z using the standard relationship for this problem. Each step is applied in turn, so every output is traceable back to the numbers you typed in.
How should I enter δx?
Use the value you actually have, in the unit shown next to the field. Enter every value in the same unit system. The calculator assumes the figures you supply are consistent, so it cannot correct a mistake that happens before the calculation starts.
Is the answer exact?
It is as exact as the formula and the inputs allow. The method itself is the standard one for this problem, so the only sources of error are the figures you enter and the precision you ask for.
Why does a small change in one value move the answer so much?
That happens when an output depends on a ratio, because dividing by a small number magnifies any change in it. Check the units first, then check whether the value you entered is in the range the relationship was designed for.