Mean of values with individual random errors

Question

I have 3 values and each of them have some random error to them. These values have no correlation with each other and neither do the errors. I want to find the mean of these values and the standard error of the mean value.

Example data:
x= 1 +- 0.03
y= 2 +- 0.54
z= 4 +- 0.22

mean= ? +- ?

I am not very experienced in statistics so please keep the answer simple to understand. I have been told that the mean is just not arithmetic mean i.e (x+y+z)/3. We need to consider the errors as some weighting.

This is what my professor told me:

To compute the mean value of the 3 independent projections, you want to use not just the values of the shear (1,2,4 in this example) but also their associated error bars. That is, you have 3 sets of data points with error bars, so you should be computing a weighted mean and deriving the final error bars using the weights in the standard fashion.

I am not sure what exactly she means by standard fashion or how to get the weighted average.

Answer 1

You could have calculated simple mean and variance and standard error as if it would just be one big stream, but calculating separately (as a weighted avg) will reduce the variance. This is a variance reduction technique that called stratified sampling.

Let's mark $n_i$ - number of numbers is stream $i$

$n$ - the total number of numbers from all stream

$p_i - \frac{n_i}{n}$ - the proportion of stream $i$

$\mu_i$ - the simple average of each stream

$\sigma_i$ - the standard deviation of each stream

$Y$ - the new variable for which we want to compute expectation and variance

Then the expectation (the weighted average) will be:

$E(Y) = \sum_{i=1}^{3} p_i \mu_i$

The variance will be:

$Var(Y) = \sum_{i=1}^{3} p_i^2 \sigma_i^2 $

$\sigma(Y) = \sqrt{(Var(Y))}$

$stdError(Y) = \frac{\sigma(Y)}{\sqrt(n)}$