*v = (0,1,2,4,8,10)*

with weights (real numbers not necessarily integers)

*w = (0,10,1,1,1,1)*

Calculating the weighted mean is obvious:

*m = sum( w_i * v_i) / sum (w_i),*

but how do I get weighted confidence interval (lets say with 95% confidence)?

The standard confidence interval formula does not work, since it is in the form

*(m- Z95 * omega , m- Z95 * omega)*, but obviously the low limit should be closer to m than the upper limit (for the data presented above).

Above omega can be chosen as the weighted standard deviation:

*D = sum( w_i * (v_i-m)^2) / sum (w_i),*

omega = sqrt(D)

omega = sqrt(D)