Let us start with random variable (x_1+x_2)/2 representing the mean from two series of observations and assuming that sigma_1 and sigma_2 defined the Gaussian distributions for the expectations.

u=(x_1+x_2)/2 obeys Gaussian distribution which is product of two Gaussians with delta function constraint forcing u to given value.

One can obtain the probability distribution for u by integrating either over x_2 or x_1 by using for instances x_1=2u-x_2. Doing the integration one obtains distribution for u which is Gaussian with

sigma= sqrt(sigma_1^2+sigma_2^2)/2.

For sigma_1=sigma_2 one obtains sigma_1/sqrt(2)

The number of sigmas for u in combined data would be

u= (n_1sigma_1+n_2sigma_2)/2 using sigma as unit giving

n=(n_1sigma_1+n_2sigma_2)/(2*sigma)

For sigma_1=sigma_2 one would have

n = (n_1+n_2)/sqrt(2)

This is the formula that also Phil wrote with s denoting n.

If sigma_1 and sigma_2 are not same, one cannot say anything definite: their ratio should be known. Sigmas characterize to some degree subjective expectations and it would be nice to know how the underlying Gaussian is deduced and can one be sure that the sigmas are same. Misunderstood something? Wise comments welcome;-)!

Thanks a lot