2
$\begingroup$

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'.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – gung - Reinstate Monica Jul 16 '13 at 0:22
  • $\begingroup$ 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'. $\endgroup$ – krokodil Jul 16 '13 at 1:15
  • $\begingroup$ In that case, @John's suggestion is probably the best that can be done. $\endgroup$ – gung - Reinstate Monica Jul 16 '13 at 1:43
1
$\begingroup$

You can use the mean of s^2 as sample variance. For sample SD you'd take the square root of that. You cannot average the standard deviations together.

While it's nice to be able to have a weighted mean by the number of samples that went into each individual variance measurement, if those aren't available then this is your next best estimate method.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – krokodil Jul 16 '13 at 1:12
  • $\begingroup$ The equations for pooling variance are just that, mean of s^2, but typically weighted. So, you can call it an equally weighted pooled variance. $\endgroup$ – John Jul 16 '13 at 3:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.