# Error propagation for non-normal errors

I have several measurements taken over a time series. Each data point has a standard error value. I need to sum up the data points, and determine the error associated with that sum. The error values across the time series are non-normal, so I'm assuming that I can't use the usual error propagation rules (i.e., SE_total = √(SE1^2 + SE2^2 +....) ). A log-transform of the errors shows that the errors are log-normal though. I'm not sure how to approach this. Is there a way to sum log-normal errors?

## Answers and Replies

mathman
Science Advisor
The formula for standard error does not require a normal distribution. The only condition is that the measurement errors be independent.

I thought that since the formula for error propagation is derived for a Gaussian distribution, the typical summation of errors in quadrature is not applicable when errors are not normally distributed.

mathman
Science Advisor
The typical derivation is based on the the following (two variables - generalizable to n):
Simplify writing by assuming means are 0. E((X+Y)²)=E(X²)+2E(XY)+E(Y²). IF X and Y are independent (uncorrelated is enough), E(XY)=E(X)E(Y)=0.

Thank you for your replies. I've done more searching on this topic and it seems I'm not alone in my confusion about this. I want to make sure I've got this right: even though the distribution of the errors is non-normal, the usual error propagation rules are applicable since the function that I'm propagating errors for is a simple sum (i.e., linear). Does the distribution of errors come into play for propagating errors for more complex, nonlinear functions?

mathman
Science Advisor
Not necessarily. E((XY)^2)=E(X^2)E(Y^2), so you have variances multiplying, not adding.

DrDu
Science Advisor
Statisticians call this the delta method, and another important assumption is that you can use a lowest order Taylor expansion. So ## Var(f(x))=\langle f^2(x)-\langle f(x)\rangle^2\rangle=\langle (f(0)+f'(0)x+f''(0)x^2/2)^2+\ldots -\langle f(0)+f'(0)x+f''(0)x^2/2+\ldots)\rangle^2 \rangle=f'(0)^2 \langle x^2 \rangle ##+ higher order terms.
The higher order terms depend not only on the statistics of x but also on the Taylor series. It might be that they all disappear for a Gaussian, as higher order correlation functions can all be expressed in terms of ##E(x^2)##.

You say that your error is lognormal distributed?
So why don't you use error propagation for the logarithmized independent variable , i.e. replacing Var(f(y) by ##Var(f(e^x))##?