Renato -

You could look for Kirk Wolter's book on variance.

 http://link.springer.com/book/10.1007%2F978-0-387-35099-8

I have a copy somewhere, but have not seen it since we moved.  :-) 

Look for something like Rhat = y/x (just reverse the roles of x and y you had), and for independence (no covariance term), you should find something like this (approximately), dropping hats here to make it easier for me to type, but using ^ to indicate an exponent: 

V(R) = V(y)/x^2  +  V(x)y^2/x^4

Cheers - Jim 

PS - Note that this works out the same, whether y and x in my sloppy notation here stand for estimated means or estimated totals.  

PSS -  Note that the standard deviation of a mean rather than the data distribution itself is called a "standard error."  Deviations refer to data distributions. 

PSSS -   :-)  I think that the theory volume (volume 2) of the Hansen, Hurwitz, and Madow book on survey sampling (circa 1952, but reprinted in the 90s) has something on the variance of ratios also.  Cochran's Sampling Techniques  (3rd ed in 1977) is excellent, but I think the only R there is in a different context than I think you want to see.

Ah! I found the relative variance of R in Cochran(1977), i.e., V(R)/R^2, at the bottom of page 183.  He references Hansen, Hurwitz, and Madow. 

Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons. 

That section goes on to talk about ratios of ratios.

Cochran refers to a theorem in chapter 2, where he notes that this is an approximation good for large n.  His context there is that of a finite population.  

 Thank you VERY much James!

Standard errors provide one measure of spread for the distribution of a random variable. If you add together a number of random variables the spread in the distribution increases, in general. For independent sums the increase is not large (and it is not just a matter of adding up standard deviations). Observe as follows:

σ2X+Y = σ2X + σ2Y  if X and Y are independent random variables

If Y = −Z, for another random variable Z, then we get:

σ2X−Z = σ2X + σ2−Z = σ2X + σ2Z  if X and Z are independent

It is positive on right-hand side as subtracting an independent quantity from X cannot decrease the spread in its distribution

So, in order to sum or substract two independent variables, could I sum their means as well as their variances?

For means:

(1/n) (µX + ... + µX ) = µX

for variances:

(1/n2) (σ2X + ... + σ2X)  by independence of the Xi

=  σ2X/n

Renato - Didn't you want variance for a ratio of independent variables, not a sum of them?  Either way, you only involve covariance when there is dependence. Regarding adding independent variable variances, that is straightforward as you noted, and is an important feature used when applying stratified sampling. - Jim 

Oh - I see - you asked about variance for  subtracting not adding means.  Yes, you still add the variances, but that is for estimated totals.  - Also, in my comment on stratification, we work with estimated totals and add them and add their estimated variances. 

Hi James - at first, I did ask about the variances of the ratios, yes, but as I forsee that I will have to use the sum of variances as well, i just followed up the answer from Prof. Robert. Thank you very much for the help!

Renato - 

In that case, I recall a general book, Probability & Statistics for Engineers & Scientists, by Walpole and Myers, that covered that sort of thing.  That was many, many years ago, and looking it up now, I see this:

 Probability & Statistics for Engineers & Scientists, 9th ed. by Walpole, Myers, Myers, Ye. 

As I recall, in my opinion, the part of the old book that covered adding and subtracting variances was good, and perhaps insightful.  That was perhaps the high point of the book.  (The first half of the book may have been best, re providing an understanding of the material.)  At least, that is my impression.  I like to know how things work. 

Hopefully you can take a look at that book or a similar one that explains this. 

Best wishes - Jim 

Howard Seltman wrote a short document of how to derive the variance of a ratio using Taylor expansion: http://www.stat.cmu.edu/~hseltman/files/ratio.pdf

You may skip the derivation and just recognize the result, which is

µ²A/µ²B * (s²A/µ²A + s²B/µ²B)

leaving out the covariance term for the case that the variables A and B are independent. Here, µ indicates the expected value (mean) and s² stands for the variance.

One can also use the E-operator ("E" for expected value). The expected value of a variable X is

µ = E(X) = integral[over the support of X] x*P(x) dx

and the variance is the expected value of the squared differences of the values from µ:

s² = V(X) = E( (X-E(X))² )

Now one can show that V(A/B) = E( ((A/B)-E(A/B))² )

what is approximately equal to E( (A/B - µA/µB)² )

what is similar to s²A/µ²B + µ²A/µ²B * s²B/µ²B

(again leaving out the covariance term)

what equals the result of the derivation through the Taylor series.

Again the same result is obtained via error-propagation (https://en.wikipedia.org/wiki/Propagation_of_uncertainty).

Dear Renato:

If you have two independent random variables, then:

E(X/Y) = E(X)·E(1/Y).

And:

V(X/Y) = E(X2/Y2) - [E(X/Y)]2 = E(X2)·E[(1/Y)2] - [E(X)·E(1/Y)]2.

I hope this would be helpful.

Dear Espejo: Thank you for the information. Very helpful here.

from Pearson (1897): http://www.ssc.wisc.edu/cde/cdewp/72-18.pdf

1) With this samle size you can bootstrap the satistics.

2) If the series A and B are paired (the i-th value of A "belongs to" the i-th value of B), then you should calculate the mean of the 1000 ratios. If A and B are independent series of repeated measurements (possibly of different lengths), then calculate the ratio of the means.

This link makes for a great cheat sheet for addition, subtraction, multiplication, division and log, which is what I needed just now.

https://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error

have a look at this paper:

Article Mean and Variance of Ratio Estimators Used in Fluorescence R...

https://pdfs.semanticscholar.org/c049/51f11ed24f4b60a73813f659578dda76bd95.pdf

Using Taylor Expansion, you can prove that an approximation to Var(r)=Var(X/Y) is r^2 * (Var(y)/y^2 + Var(x)/x^2 - 2*Cov(x,y)/xy), where r, x, and y are the means of their distributions. See page 37 of https://www.sagepub.com/sites/default/files/upm-binaries/6427_Chapter_4__Lee_(Analyzing)_I_PDF_6.pdf

What the text does not discuss is that this approximation is great (

have a look at this paper: https://pdfs.semanticscholar.org/c049/51f11ed24f4b60a73813f659578dda76bd95.pdf