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

Question

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?

Answer 1

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 $\Delta$.

This is a fairly common method to assess the quality of a model if you don't have error bars on the data. After all, we do not know how accurate the data in the table is. In the limit of infinitely large error bars any model would be good.

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.)