# What is the N after propagation of errors?

#### RyanZheng

##### New Member
Say X = (A + B)/2; A is measured 100 times and B is measured 100 times, so we get the mean +/- SD for both A and B. Through propagation of uncertainty, we can also get the mean and standard deviation of X. But what is the N associated with that SD? Is it 2 or 100 or 200?

I am asking this because I like to see if X and Y have significant differences, when X = (A + B)/2 and Y = (C + D)/2; A, B, C, and D were measured 100 times each. I will need the mean, SD, and N for X and Y for a t-test between X and Y.

Thanks,

#### katxt

##### Well-Known Member
Not 2.
And it makes little practical difference whether you use 100, 200 or 400 for the final test between X and Y.
I presume that when you do your error propagation, by SD you mean the SE, the standard error of the mean, rather than the SD of the 100 numbers.

#### RyanZheng

##### New Member
Thanks. I was actually using SD and thought that error propagation works for both SD and SE. Correct me if I am wrong. Any insight on this will be helpful.

#### Dason

I mean it sounds like at the end of the day you have 100 observations from X and 100 observations from Y. So - you really could just ignore how you got to X and Y since those are the variables of interest here. So you have a t-test with 100 observations from X against 100 observations from Y.

#### RyanZheng

##### New Member
It is not that easy I think. Because A and B are not paired, so I don’t get simple 100 observations of X. X is simply the average of A and B, which are independently measured.

#### katxt

##### Well-Known Member
Thanks. I was actually using SD and thought that error propagation works for both SD and SE. Correct me if I am wrong. Any insight on this will be helpful.
It sorta does but in this case, your aim is the uncertainty of Y-X so SE is the thing.
My experience of error propagation goes like this.
Find the SE of the means A and B (=SD/sqrt(n)). These give the uncertainties in A and B or SEa and SEb. This is where the n comes in.
Find A+B and the uncertainty in A+B = SEa+b =sqrt(SEa^2+SEb^2) (the notation is a bit wobbly but you'll get the idea. I have always used a table layout.)
Find X and the uncertainty in X = SEx = 1/2 the uncertainty in A+B = 1/2.SEa+b
Repeat for the uncertainty in Y SEy.
Find Y-X and the uncertainty in Y-X = SEy-x =sqrt(SEx^2+SEy^2)
The value of Y-X is tested against 0 using SEy-x.
Strictly this test should use the df of Y-X which I suspect is 399, but it wouldn't matter much if you used 100, or 200, or 400 df or used a z test, or just used 2*SEy-x to get a 95% CI on Y-X. Then ask does 0 lie within 2*SEy-x of the difference?