@Raspi Rascal

Yes, I am enjoying this discussion. And for me, it is timely. My job is requiring me to do more and more statistics.

Keep up the good work.

@MichaelHelmle did give you enough information to determine the answer for WeightedData. And it is readily available at a Wiki page (and elsewhere).

Using Michael's notation the weighted mean is given by

$$\bar{v}=\frac{\sum _{i=1}^n v_i w_i}{\sum _{i=1}^n w_i}$$

And the weighted variance is given by

$$\frac{\sum _{i=1}^n w_i (v_i-\bar{v})^2}{V_1-\frac{V_2}{V_1}}$$

where

$$V_i=\sum_{j=1}^n w_j^i$$

The weights are assumed to be known and not random variables.

Since Mathematica can do many caclulations symbolic, you can build a small symbolic weighted data list and use this:

wdata2 = WeightedData[Array[v, 3], Array[w, 3]]

Here you get the formula how Mean and Variance are calculated

Mean[wdata2]
Variance[wdata2]

Thanks but that's exactly the point which i was making. Your output shows the end result, not the definition. I cannot verify where the output comes from. You will not find such an output as the literal definition of the variance, in any reference book, textbook, or elsewhere.

I shouldn't give up here.

The topic is relevant in practice. Knowing what you're doing is relevant in practice ... because weighted data is everywhere (including introtexts) and the calculation of variance is everywhere too. So if you're not sure what you're doing, you will end up producing the wrong variance value for your problem. Thanks to the documentation of Mathematica :P