# The difference between a standard error and a propagated error

I'm confused about the difference between the standard error and the propagated error. For example, say that I got two measurements which I want to average: $$16.04 \pm 0.31$$ and $$15.72 \pm 0.28$$. The average of these two measurements is 15.88 and I wonder what is the error of this average value. To my knowledge, it seems that there are two ways of calculating the error as shown below.

• Method 1: $$\sqrt{\frac{0.31^2+0.28^2}{2}}=0.30$$ I'm not sure if there is a terminology for this value. I don't think it's a standard error, which should be $$\sigma/\sqrt{n}$$, although it looks pretty similar. This value looks pretty reasonable in my case. This method is used here.
• Method 2: According to the error propagation formula, the error of $$z=(x+y)/2$$ is $$\frac{1}{2}\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}}$$. In this case, it would be 0.21, which seems unreasonable in my case.

I'm wondering the difference between these two methods and when I should use which.

• Average is simply 0.5 * (x + y). SD is then 0.5 * Sqrt(Var(x + y)), which leads to the second case. Also, the second case is what follows from the page you linked to, not the first. Apr 3 at 4:33