# How to get an overall standard error from weighted combination of standard errors based on same sample but different statistics

Trying to reformulate my question.

Let's say i have three statistics from a sample. I have their variances $$s_1, s_2, s_3$$ and standard errors of the mean $$se_1, se_2, se_3$$. They are all based on sample size $$n$$, all from the same population and same sample.

I have a metric based on a weighted combination of each of these statistics. I want to combine them to get an overall score with a standard error of the mean for the score.

So the combination is something like:

$$Score = stat_1 + 20\cdot stat_2 + 50 \cdot stat_3$$

I can calculate a variance for the score just by using the sample variance addition formula.

But how can I get a standard error of the mean for the score based on the information I have? Do I need to just sample the score directly or can I somehow combine the sample's standard errors or variances to get a standard error of the mean for the score?

In general you cannot calculate the variance of the combination by the sum of the variances. It is true that for independent random variables, $$V[X + Y] = V[X] + V[Y]$$. But because these statistics are all coming from the same sample the assumption that they are all independent is highly questionable. Instead you also need to consider the covariances between them.

What you need is the full variance-covariance matrix of the sample statistics. IE, if your statistics are $$\bar{x},\bar{y},\bar{z}$$ then we can write the covariance matrix as

$$V[(\bar{x},\bar{y},\bar{z})^\prime] = \begin{bmatrix} \sigma_{\bar{x}}^2 & \sigma_{\bar{x}\bar{y}} & \sigma_{\bar{x}\bar{z}} \\ \sigma_{\bar{y}\bar{x}} & \sigma_{\bar{y}}^2 & \sigma_{\bar{y}\bar{z}}\\ \sigma_{\bar{z}\bar{x}} & \sigma_{\bar{z}\bar{y}} & \sigma_{\bar{z}}^2 \end{bmatrix} = \Sigma$$

Then the variance of the combination can then be found using standard algebraic operations

$$V[(1, 20, 50)(\bar{x},\bar{y},\bar{z})^\prime] = (1, 20, 50) \Sigma (1, 20, 50)^\prime$$

The standard error of the combination is the square root of that number.

• this is a beautiful reply, thank you! Do you know how I can get the covariance of the variances in the case where they are standard error estimates from three different models? Commented Dec 21, 2022 at 0:14
• I added a question here: stats.stackexchange.com/questions/599682/… Commented Dec 21, 2022 at 4:13