# How to find the standard error of calculated results in comparison with experimental data

Given two sets of data, experimental frequencies and calculated frequencies from some theoretical model: In an article underneath that table, one can find the statement that the "standard error for all calculated values from experimental values is: $$\Delta = 2.74$$".

The author obviously wanted to test their theoretical model against experimental data, but how do they get $$\Delta$$?

Generally, the question is: if I discover a new theoretical model, and calculate freqeuencies according to that new theory, how do I find the (standard) error/deviation $$\Delta$$ of my calculated results?

In this case they are referring to the root-mean-square deviation, that is, if you have a set of $$N$$ measurements $$y_i$$ and a model $$y(x)$$:
$$\Delta = \sqrt{\frac{\sum\limits_{i=1}^N \left(y_\mathrm{i} - y(x_i) \right)^2}{N}}.$$
If I evaluate that for the data set you posted I get your value of $$\Delta$$.
In general it is best to look at the reduced $$\chi$$-squared, since this accounts for both the error bar and the number of parameters used in the model. (Obviously, a model that uses more parameters and is as good as a model that uses fewer, is worse.)
• Thank you. Actually that was the first thing I tried. But somehow I used $N=16$ instead $N=12$ :/ – multipole Apr 11 '19 at 13:51