# How to calculate the uncertainty and mean of multiple measurements with different errors?

In common situation,each measurement with the same error,so we can simple average the measurement value to get the result and uncertainty.But some times the error of each measurement is not same,that may beacause the instrument retrieval of indirect Physical quantity(such from the optical to concentration). So we get a series of data

$$x_i$$ with error $$e_i$$ i~1,2,3,..,N.

So,how to represent the result and the uncertainty by ($$\hat{x} \pm \hat{\sigma}$$)?

Consider a series of $$i = 1, 2, ..., k$$ measurements each with value $$x_i$$ and standard deviation $$\sigma_i$$.

The best estimate for the overall mean is $$m_{best} = {\sum_{i = 1}^{k} x_i/\sigma_i^2 \over \sum_{i = 1}^{k} {1 \over \sigma_i^2}}$$ and the best estimate for the standard deviation of the mean is $$S_{best} = ({\sum_{i = 1}^{k} {1 \over \sigma_i^2}})^{-1/2}$$. You report $$m_{best} \pm S_{best}$$ for your final result.

Note: The mean $$m_{best}$$ and standard deviation of the mean $$S_{best}$$ are best-estimates for the unknown population values mean $$\mu$$ and standard deviation of the mean $$\sigma_{\mu}$$.

(For example, see the text Data Analysis for Scientists and Engineers by Meyer for details.)

More generally, you may have a series of samples to combine, each sample with a set of more than one measurement. For this case, see my answer to Uncertainty in ripetitive measurements on this exchange.

• Thank you for your answer! Commented Nov 30, 2021 at 5:19
• You are welcome. Commented Nov 30, 2021 at 13:01
• Great answer! It helps to realise that that the average is a weighted average with weights $w_i \propto 1/\sigma_i^2$. In addition, the formula from the standard deviation can be motivated by reading about Baysian statistics. In fact, this is the standard formula which is used to weight the "prior knowledge" and "new data" to obtain the "new state of knowledge". Commented Nov 30, 2021 at 19:53
• Thank you @Semoi. You may be interested in my rambling answer to physics.stackexchange.com/questions/675632/…, particularly related to belief/plausibility. Commented Nov 30, 2021 at 21:22