Why is the error value higher in the average calculation?

[davidp92]

Not sure if this is the right section to post this..
I have 3 measurements and was trying to take the average of the measurements and calculate the error of the average:
replicate 1 = 8.9 (+/-) 0.71mg
replicate 2 = 9.3 (+/-) 0.69mg
replicate 3 = 8.8 (+/-) 0.70mg

I get an average of 8.9333 (+/-) e where e=sqrt((rep 1 error)^2 + (rep 2 error)^2 + (rep 3 error)^2) which gives me a value of 1.21. But why is the error value so much higher in the average?
What step am I missing? I don't know the derivation behind the error propagation formula - so I just use it as it is: e=sqrt((e1)^2+(e2)^2+...)

 

[Delphi51]

When I was in first year physics, they told me to divide the error by the square root of the number of measurements I had. This is contrary to the usual rules of error propagation. I found it in one place
http://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm
Look at section 3.10 near the bottom of the page.
Unfortunately it is very difficult to read due to typos and anyway, I think they are just saying there is a rule somewhere else that you divide by root n. It certainly makes sense that your average should be more accurate than each measurement is.

EDIT: more on how it is derived here: http://en.wikipedia.org/wiki/Standard_error_(statistics)