Weighted mean from data Suppose we have 3 experimental groups measuring "x", they measure
$$x_1=\mu_1\pm \sigma_1$$
$$x_2=\mu_2\pm \sigma_2$$
$$x_3=\mu_3\pm \sigma_3$$
In the 3 experiments, we have done N measurements of "x". We do know that we have the best estimate of "x" weighting the three means or averages $\mu_i$ with the variance. But, suppose we provide a statistical "weight" (probability or reliability) of those 3 values, e.g. 0.25, 0.50 and 0.25 to the first, second and third. More generally, we give weights or probabilities $w_i$ to those calculations. How can we calculate the right weighted mean including both, variance and the probabilistic weights $w_i$? How does the solution change if we have different sample sizes in the 3 different experiments, i.e., how is the solution modified when we have sizes $N_1, N_2, N_3$ in the 3 experiments instead of the same number N of observations/measurements?
Remark: I have written the error in the mean/average as $\sigma_i=s_i/\sqrt{N}$
 A: The weighted mean of three values is given by
$$ \bar{x} = \frac{ \sum_{i=1}^{3} x_i \alpha_i}{\sum_{i=1}^{3} \alpha_i},$$
where here $\alpha_i$ represents the weight that you give to each measurement.
If you wished to just find the weighted mean from your data as you presented it originally, then the weight $\alpha_i = 1/\sigma_i^{2}$.
If you then wish to insert additional arbitrary weightings, $w_i$ to each point then the weights will multiply each other. i.e. $\alpha_i = w_i/\sigma_i^{2}$.
Here I am making the assumption that this is what you mean by a "weight" and not that there is some extra systematic uncertainty in the measurements that is shared in the proportions you suggest between the three measurements. If that were the case then you would form the weight by adding the appropriate systematic error in quadrature with $\sigma_i$.
As to the last part of your question I am a bit puzzled. If you are using the fact that $\sigma_i = s_i/\sqrt{N}$ (I assume $s_i$ is the standard deviation of the population right?), then the possibility of different sample sizes is already incorporated into this formula - the experiment that take a larger number of measurements will presumably return a smaller value of $\sigma$ (if $s$ stays the same).
A: The way that I would approach this is to calculate $\chi^2$ with
$$\chi^2 = \sum_{i=1}^N\left({p - x_i \over \sigma_i}\right)^2$$
where $p$ is the best value for the fit and $N$ is the number of measurements $x_i$ with error $\sigma_i$. 
Now with a computer or Excel sheet I would plot $\chi^2$ as a function of $p$.
As you know the value of $p$ for the lowest value of $\chi^2$, which we refer to as ${\chi_{min}^2}$, is the best fit.
To get the uncertainty you look to see the range of $p$ values for which 
$$\chi^2 <{\chi_{min}^2}  + 1$$  
This will give you the range for your error bars for $p$.
This is a good general method for fitting data, but it gets more complicated when you fit more than one parameter and parameters can depend on each other. A good place for more information about this is the Numerical Recipes Book by Press, Teukolsky $et al.$ and here on the web
