# Thread: Standard deviation of the Ratio of two means

1. ## Standard deviation of the Ratio of two means

Hi, guys

I have two sets of data: A and B.

I want to calculate a ratio of average(A)/average(B).

I wonder if there is a way to get the standard deviation of this ratio?

thanks.

2. ## Re: Standard deviation of the Ratio of two means

Suppose avg(A) and avg(B) are normally distributed. This will be exactly true if A and B are normally distributed, because a sum of normally distributed deviates is itself normally distributed. It will be approximately true for large samples however A and B are distributed, by the central limit theorem.

Then what you are asking is how the ratio of two independent normal deviates is distributed. The answer is the Cauchy distribution. (http://en.wikipedia.org/wiki/Normal_..._distributions)

The Cauchy distribution has infinite variance, so your ratio's standard deviation is infinite.

3. ## Re: Standard deviation of the Ratio of two means

http://en.wikipedia.org/wiki/Taylor_...ndom_variables

You may try to replace the theoretical moments by the sample moments, provided that
you have collected the sample means, sample variances and the sample covariance of the
two data sets.

4. ## Re: Standard deviation of the Ratio of two means

Originally Posted by BGM
http://en.wikipedia.org/wiki/Taylor_...ndom_variables

You may try to replace the theoretical moments by the sample moments, provided that
you have collected the sample means, sample variances and the sample covariance of the
two data sets.
Thanks, this may work.

Just one question, in the equation E[Y]^2, does it mean E([Y]^2) or (E[Y])^2?

5. ## Re: Standard deviation of the Ratio of two means

Using Taylor Expansion, you can easily see the result:

Let

Then

Taylor Expand at , we have

6. ## Re: Standard deviation of the Ratio of two means

Originally Posted by BGM
Using Taylor Expansion, you can easily see the result:

Let

Then

Taylor Expand at , we have

Thanks, I seem to get it.

For my data, A and B are independent, does it mean that Cov(A,B)=0?

7. ## Re: Standard deviation of the Ratio of two means

Yes if you accept that the independent assumption.

Reminder: As what Ichbin mentioned above, the variance of the ratio is usually
large if the random variable in the denominator have a high probability to
fall in the neighborhood of 0. So make sure, if you have make any distributional
assumptions, check whether the moment exist or not.

8. ## Re: Standard deviation of the Ratio of two means

that ratio should follow the ratio normal dist. See this paper: D. V. Hinkley (December 1969). "On the Ratio of Two Correlated Normal Random Variables". Biometrika 56 (3): 635–639 and substitute the corresponding parameters. If the means are zero you should Cauchy dist even the variances of the A and B are not one. Also Cauchy dist is a special case of the ratio normal dist.

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