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.
In that case, the weightedaverage of one value is (W1*M1) divided by the sum of the weights.
If someone were to weight the shot by shot length the weightedaverage would have been: (10.3*10.3 + 10.8*10.8 + 10.5*10.5)/(10.3+10.8+10.5) or about 10.54 While shot length may, in some cases, be a good estimate of the correct weights, one can see here that it is not the same as using the correct weights.
Weight matrices on physical grounds have to be symmetric, so the weight has to have N columns and N rows.
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