Math Question Related to Adding Error in Quadrature

Factual Questions
1

Today, when I got my homework back from my Experimental Physics Class, I was marked off on this type of question.

X=3Y+Z+T

I said:
X=Y+Y+Y+Z+T

error(X)=(error(Y)^2+error(Y)^2+error(Y)^2+error(Z)^2+error(T)^2)^(1/2)
=(3(error(Y)^2)+error(Z)^2+error(T)^2)^(1/2)

But, the prof. was adamant that this was incorrect (even though it followed the traditional rules of algebra) and instead that this was correct:
=((3error(Y))^2+…)^(1/2)

The reason given by the prof was that the errors should not be added like regular terms but simply squared right away, although he himself had never fully understood the reason. He demonstrated another argument through derivatives that I was unable to follow, and he referred me to another professor who might be better able to explain it. Since I won’t have an opportunity to reach this prof. until Friday, Dopers?

2

Here’s a simple way of thinking about it logically: The error in 3Y logically has to be thrice the error in Y, so

[symbol]s[/symbol][sub]3y[/sub] = 3[symbol]s[/symbol][sub]y[/sub].

If we add things your way, we get

[symbol]s[/symbol][sub]3y[/sub] = sqrt(3) [symbol]s[/symbol][sub]y[/sub].
Basically, what’s happening is that when we do things your way, what you’re tacitly assuming is that each measurement in Y is independent, so the errors needn’t be identical and in the same direction. But of course they are identical and in the same direction, because it’s only one measurement. So you’re understating the uncertainty.

3

g8rguy,

You’re a gr8 guy. That makes perfect sense, I had just never realized that. The error is always the same error. You saved me an hour and a half with a professor that likes no-one and never bothers to show up to his class anyway (literally have not seen him once, but I hear he doesn’t like anyone).

Cool.