To estimate variance (or the mean for that matter) of some distribution you'll need something stronger to justify the "standardized weights" as probabilities. Was this a "simple random sample" from a larger population of integer values?

The formula that seems to be used for Variance with weighted data is as follows:

Variance[WeightedData[x, w]]
(Total[w x^2] - Total[w x]^2/Total[w])/(Total[w] - Total[w^2]/Total[w])

This (as you've noticed) doesn't always match what one would expect when the weights are frequency counts. Here are two examples. When the weights are all 1, then all formulas match:

x = {1, 12, 3};
w = {1, 1, 1};
Variance[WeightedData[x, w]]
(* 103/3 *)
(Total[w x^2] - Total[w x]^2/Total[w])/(Total[w] - Total[w^2]/Total[w])
(* 103/3 *)
Variance[{1, 12, 3}]
(* 103/3 *)
(Total[w x^2] - Total[w x]^2/Total[w])/(Total[w] - 1)
(* 103/3 *)

But if the weights are changed from 1's, then Variance with weighted data doesn't match:

x = {1, 12, 3};
w = {2, 2, 2};
Variance[WeightedData[x, w]]
(* 103/3 *)
(Total[w x^2] - Total[w x]^2/Total[w])/(Total[w] - Total[w^2]/Total[w])
(* 103/3 *)
Variance[{1, 1, 12, 12, 3, 3}]
(* 412/15 *)
(Total[w x^2] - Total[w x]^2/Total[w])/(Total[w] - 1)
(* 412/15 *)

I, too, don't know where they got their formula for variance for weighted data. At minimum, it would seem to call for a more specific reference in the Documentation Center. If values are what one expects only when the weights are 1, then that seems to defeat the purpose of weighted data.

Hello! I suppose that's a correction for bias.

Suppose the weight is uniform. Then Dot[weights, weights]=1/n and const=n/(n-1). But, in the general case,

const = 1 / (1 + -Dot[weights, weights])

seems different from the standard coefficient

https://en.wikipedia.org/wiki/Weightedarithmeticmean#Reliability_weights

which should be the inverse of

1 - (Dot[weights^2, weights^2]/Dot[weights, weights]^2)

For instance

  pds = Table[1/p, {p}];
    1 - ((Dot[pds^2, pds^2]/Dot[pds, pds]^2))

gives (p-1)/p. Maybe there are elementary alterations of the weights (normalization, etc.)?

After some digging in to the internal code I found out what the factor is. It is given by:

const = 1 / (1 + -Dot[weights, weights])

where weights are now normalized weights = weights/Total[weights] which works out to be 1.27277 in my case and that exactly explains why Variance gives a higher value. The origin of this factor is however unclear to me... anyone?