Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counter-intuitive properties, as captured for instance in Simpson's paradox.
Arithmeticmean - the sum of all measurements divided by the number of observations in the data set
Weightedmean - an arithmeticmean that incorporates weighting to certain data elements
The only significant reason why the arithmeticmean (classical average) is generally used in scientific papers is that there are various (statistical) tests which can be applied to test the statistical significance of the results, as well as the correlations that are explored through these metrics.
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