# Estimating population variance considering both population sample variance AND sampling method variance

I want to estimate the variance of a normally distributed population. I can take N samples and calculate the sample mean and sample variance, which would normally suffice; however, the sampling method itself has inherent variability that is known. Is there a way to "subtract out" the known test method variance the get a better estimate of the population variance?

To provide some context, I'm measuring density of an asphalt road after construction. Random locations are selected for measuring density for quality assurance. There are two methods, A and B, for measuring density and they have different known variance. The method variance was found by repeated measurements on known references. If I use Method A, with high variance, to and get N samples of the road density, I'll end up with a sample variance that is higher than if I used the more precise Method B. And in both cases, my variance will still be higher than the true variance of my sample. If I know the actual test method variance, isn't there some way to leverage that information when making an estimate of the sample and population variance?

• Commented Jun 28, 2019 at 14:05
• Can you explain how that would apply to my situation? I'm not sure I follow. Commented Jun 28, 2019 at 20:47
• The sentence makes sense to me. What's the problem you see with it? Commented Jun 29, 2019 at 7:17
• The best estimate of the population mean is going to be the sample mean. I am not sure what you are asking. Commented Jun 29, 2019 at 12:37
• Thanks for the comments. I see I've been asking the wrong question, so I've rewritten the question to focus on estimating the variance. Please see the revised question. Commented Jun 30, 2019 at 3:36

I imagine that you use the testing method to take many measurements and get what looks like a normal distribution of Variance 5 and mean 10. You then want to update your understanding based on the documentation for the test method, which says that for any samples of known density x, the testing method's measurements will have a normal distribution with a variance of 3 and a mean of x+2. From $$Var(X+Y) = Var(X) + Var(Y)$$, you can then estimate that your road locations do not all have exactly the same density, and instead have densities normally distributed with Variation 2 and mean 8.