Error propagation

[Biochemistica]

Hi,

 

I have a number of measured data consisting of 10 numerical values. Let’s call these x1 to x10. x1 to x10 have a certain standard deviation, which I calculated by using the formula for the standard deviation for samples (see for example here: https://statistics.laerd.com/statistical-guides/measures-of-spread-standard-deviation.php).

 

Then, I need to convert these values to new values according to the following formula: 2^(-x)

 

From this, I get 10 new numerical values. I can calculate the new sample standard deviation for these based on the formula above or I can use the rules for the Gaussian error propagation to calculate the new standard deviation (see attachment, here I named the new error delta(z)).

 

I get different results using one or the other method. To illustrate my calculations, I also attached an Excel table (zipped file). Which standard deviation is correct and why?

Error Propagation.zip

Edited August 30, 2016 by Biochemistica

[Klaynos]

What is the distribution of the data?

 

What can you say about your sample size?

 

I'm a big fan of error analysis, to the point where I have two papers that rely heavily on it and used to teach it to physics undergraduates.

 

I'd highly recommend introduction to error analysis by John R Taylor.

[Biochemistica]

Thanks for the book recommendation. Can you also point me to the papers that you mentioned? My university has many online journal subscriptions, so I might be able to get easy access to them.

 

The original data are approximately normally distributed.The converted data (i.e. 2^(-x)) are not.

 

My sample size is 10, which is not very large but cannot really be increased much more due to limitations in the biological material necessary for the measurements.

As a side note: When I have data sets that are not independent (e.g. paired data), I have to use error propagation formulae that take into account the covariances. Using these formulae, I get the same result for the new "error" based on error propagation and based on calculations with the individual values. This makes sense to me. But I am confused by the fact that for the mathematical conversion I get two different results depending on which approach I use.