# Error propagation for non-normal errors

• SamanthaYellow
In summary: The error propagation rule is based on the assumption that the errors are Gaussian distributed. But since the distribution of the errors is not Gaussian, the error propagation rule does not work.

#### SamanthaYellow

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?

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.

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?

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

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))##?

## 1. What is error propagation for non-normal errors?

Error propagation for non-normal errors is a statistical method used to estimate the uncertainty or error in a numerical result when the underlying data does not follow a normal distribution. This method takes into account the non-normality of the data and provides a more accurate estimation of the error compared to traditional methods that assume normality.

## 2. Why is error propagation important for non-normal errors?

Error propagation is important for non-normal errors because it allows for a more accurate and reliable estimation of the uncertainty in a numerical result. When the underlying data is not normally distributed, traditional methods may underestimate or overestimate the error, leading to incorrect conclusions or decisions based on the results.

## 3. How is error propagation for non-normal errors different from traditional error propagation?

Error propagation for non-normal errors takes into account the non-normality of the data, while traditional error propagation assumes that the data follows a normal distribution. This means that the calculations and assumptions used in non-normal error propagation are different from those used in traditional methods.

## 4. What are some common non-normal distributions that require error propagation?

Some common non-normal distributions that require error propagation include Poisson, exponential, and log-normal distributions. These distributions are commonly encountered in scientific and engineering data, and their non-normality can greatly affect the accuracy of error estimation.

## 5. Can error propagation be applied to any type of non-normal distribution?

Yes, error propagation can be applied to any type of non-normal distribution as long as the distribution is known or can be estimated accurately. However, the specific method and assumptions used for error propagation may vary depending on the type of non-normal distribution.