How to Calculate Uncertainty in a Function

Here, the uncertainty would be a measure of how close each dart comes to the average of the three dart positions.

In experimental analysis, the uncertainty is a statistically determined margin of error in associated measurements. The initial uncertainty will usually take the form of the statistical mean's standard deviation, or the standard deviation of the mean. To calculate the uncertainty of a function whose dependent variable is a statistical mean, the rules of error propagation must be followed in determining the correct contribution(s) to the overall uncertainty. Each contribution is respective to the mean from which the initial uncertainty originates.

Instructions

Example: F = x^2 - 6x

1. Write down the function whose uncertainty is to be determined. Here this appears as "F = x^2 - 6x."

2. Write down the mean value for the "x" quantity. The mean M of x would simply be the sum of all measurements xi of x divided by the total number of measurements N: M = sum(xi, i = 1 . . . N)/N.

3. Write down the standard deviation SD of x. This value is equal to the square root of the quantity of the sum of the squares of the differences between each individual measurement of x, xi, and the mean of x, M, divided by the total number of x measurements minus one: SD = sqrt(sum((xi - M)^2, i = 1 . . . N))/(N - 1)). If the standard error SE is the uncertainty that you would like to use in your calculation, simply divide the standard deviation by the square root of the total number of measurements: SE = SD/sqrt(N).

4. Take the first derivative of F with respect to x, dF/dx. Here, this appears as "dF/dx = 2x - 6."

5. Multiply the first derivative of F by the standard deviation (or the standard error) to get the overall uncertainty in F with respect to x (UFx). Here, this appears as UFx = dF/dx(SD) = (2x - 6)(SD).

References

• Introductory Physics Labs: Case Western Reserve University - Appendix V. Uncertainties and Error Propagation