# How to combine variances from sensors where each observation has its own variance?

I have a set of measurements $x_1$ ... $x_n$. These measurements are normally distributed, measuring the same value. However due to the way the data is measured, each $x$ has its own standard deviation: $s_1$ ... $s_n$. In other words I have a sensor which returns pairs (x,s). Now, I wan to estimate parameters of distribution of $x$ using $\text{N}$ samples. The common mean would be just a sample mean. What about variance? I could not apply "pooled variance" formula because I do not know how many samples were used to estimate each $s$.
Can I just use a mean of $s$ as sample variance?

Update: I can not make any assumptions on how sensor produced (x,s) values. They might be based on some hidden iterations or perhaps something else. However it is safe to assume that each (x,s) pair returned to me is independent from others and measuring the same true value of 'x'.

• Some questions: The idea here is that the sensor actually took a number of measurements, and calculated & returned their mean & SD, is that right? Do you know if the measurements from iteration to the next are independent? What is it that you want to do with these measurements in the end? You may also find this thread interesting: algebra-for-data-confidence. – gung - Reinstate Monica Jul 16 '13 at 0:22
• I could not make any assumptions on how sensor produced (x,s) values. They might be based on some hidden iterations or perhaps something else. However it is safe to assume that each (x,s) pair returned to me is independent from others and measuring the same true value of 'x'. – krokodil Jul 16 '13 at 1:15
• In that case, @John's suggestion is probably the best that can be done. – gung - Reinstate Monica Jul 16 '13 at 1:43

• Thanks! I was wondering if there is a formal name for this (using mean of $s^2$) or perhaps it could be derived from some existing ones. – krokodil Jul 16 '13 at 1:12