# Formula for pooling means, variances

We are performing an inter-comparison experiment for three devices that can measure concentration (cells/mL). We take one solution and run it through each device, and we compute mean concentration and variance on that device. I would like to find the average of the means and weight it according to sample size and variance.

I found an example applet which illustrates calculation, but cannot determine the source code formula:

https://home.ubalt.edu/ntsbarsh/business-stat/otherapplets/Pooled.htm

Given sample size, variance, and mean what is the formula for pooling these measurements to obtain an ensemble mean?

Is there a better approach?

• "Fraction of the mean of all the device means" already is a clear, specific formula. Notice that it doesn't mention variances, either. – whuber Feb 1 '18 at 19:32
• Thanks. What I really want is a way to weight the means according to variance and samples size. I edited the question to be: "I would like to find the average of the means and weight it according to sample size and variance." – Noah Feb 1 '18 at 23:08
• That, too, is a perfectly clear and computationally effective formula. But I suspect you want to weight the means by the reciprocals of their (estimated) variances, not by the variances themselves. – whuber Feb 1 '18 at 23:22
• Thanks! Can you provide a URL, citation, etc. where I can find the formula? – Noah Feb 2 '18 at 16:44
• This is standard, so expect to find an account of it in any textbook on regression, for instance. For the justification see stats.stackexchange.com/questions/12251 and perhaps stats.stackexchange.com/questions/65484. – whuber Feb 2 '18 at 17:11