Mean of values with individual random errors

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.

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:

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

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

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

• I am a little confused what you mean by stream! I just have 3 values and 3 error values associated to them. Sorry if my question is naïve! Oct 3, 2021 at 0:03
• @AryanBansal hmm so i don't understand... how can a single value has standard error? i thought it has a stream of values Oct 3, 2021 at 9:40
• I guess it is more like averaged error of the 3 errors. Mean = ? +- ?. Oct 4, 2021 at 14:23
• if the value has standard error it means that it has also standard deviation and many samples.... can you post the original question from the book (or any other source?) Oct 4, 2021 at 14:35
• It is not a question from a book. It is from my research work, we have some simulations in different projections: xy, yz and xz of some galaxies. We need to find something called circular shear stress which is given by the average of these 3 projections. Each projection has a random error to it and thus the circular shear i.e mean of these also has some standard error! Oct 5, 2021 at 0:49