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  1. I have multiplied together two means and now want to calculate the overall standard deviation. The two means and standard deviation are here: 13.7 +/- 12.7 (1SD) and 4.0 +/- 2.6 (1SD). So the answer would be 55.11 with what overall standard deviation?

  2. I then want to multiply this answer of 55.1 by 92. There is no standard deviation attached to this because it is 92 days. Does the standard deviation stay the same as the answer for question 1?

Edit: My overall goal is looking at seed dispersal in the dry season. There are around 13 seeds in a poo and they poo around 4 times a day. That tells me the number of seeds dispersed on average per day. By multiplying that by 92 it gives a rough indication of the number of seeds dispersed across the dry season. This hasn't previously been studied before, it's not designed to be overly accurate but demonstrate the importance of mutualism in seed dispersal.

Katharine Davies
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    Why do you do this? You should state your goal. You want to estimate the product of two means $\mu_1 \mu_2$ by multiplying the two sepoarate means $\bar{y}_1 \bar{y}_2$, and then want the standard error of this estimator? – kjetil b halvorsen Aug 01 '17 at 09:31
  • I am trying to think of a situation where this would be a good idea and I can't come up with one. Why are you doing this? The mean of a product can be calculated from the data. Why do you want the product of the means? – Peter Flom Aug 01 '17 at 12:21
  • You might find it handy to search this site for "error propagation" and see whether the techniques outlined there help you with your situation. – mdewey Aug 01 '17 at 12:58
  • My overall goal is looking at seed dispersal in the dry season. There are around 13 seeds in a poo and they poo around 4 times a day. That tells me the number of seeds dispersed on average per day. By multiplying that by 92 it gives a rough indication of the number of seeds dispersed across the dry season. This hasn't previously been studied before, it's not designed to be overly accurate but demonstrate the importance of mutualism in seed dispersal. – Katharine Davies Aug 01 '17 at 15:59
  • Could you please add this new information to the original post? – kjetil b halvorsen Aug 01 '17 at 16:13
  • Try this link http://www.fas.harvard.edu/~scphys/nsta/error_propagation.pdf and see if it helps. – mdewey Aug 01 '17 at 16:20
  • Thank you mdewey, number 3 on that link answers the part 2 of the question! Still stuck on part 1 though. – Katharine Davies Aug 02 '17 at 08:58

2 Answers2

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I found some old notes on this which I think come from the page I linked to in comments above http://www.fas.harvard.edu/~scphys/nsta/error_propagation.pdf so any intellectual credit goes to the authors of that set of notes. The first question of the OP is answered I believe in the section Multiplication (or division) and the second in the section Multiplication by a constant.

If you measure some quantities $a, b, c, \dots$ with uncertainties $\delta{a}, \delta{b}, \delta{c}, \dots$ what happens when you combine them?

Addition (or subtraction)

If \begin{equation} Q = a + b + \dots + c - (x + y + \dots + z) \end{equation} then \begin{equation} \delta{Q} = \sqrt{(\delta{a})^2 + (\delta{b})^2 + \dots + (\delta{c})^2 + (\delta{x})^2 + (\delta{y})^2 + \dots + (\delta{z})^2} \end{equation}

Multiplication (or division)

If \begin{equation} Q = \frac{ab\dots{}c}{xy\dots{}z} \end{equation} then \begin{equation} \frac{\delta{Q}}{\vert{}Q\vert} = \sqrt{\left(\frac{\delta{a}}{a}\right)^2 + \left(\frac{\delta{b}}{b}\right)^2 + \dots + \left(\frac{\delta{c}}{c}\right)^2 + \left(\frac{\delta{x}}{x}\right)^2 + \left(\frac{\delta{y}}{y}\right)^2 + \dots + \left(\frac{\delta{z}}{z}\right)^2} \end{equation}

Multiplication by a constant

If $A$ is known exactly and \begin{equation} Q = Ax \end{equation} then \begin{equation} \delta{Q} = \vert{}A\vert\delta{x} \end{equation}

mdewey
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    Are the quantities $a, b, \ldots$ etc. assumed to be uncorrelated or independent? If not, the $\delta Q$ for addition/subtraction needs to have the sum $\sum$ of the covariances terms added inside the square root. For the multiplication/division how does $\delta Q / |Q|$ help with the OP's question for a confidence interval? Also, do the quantities measured have to be uncorrelated/independent? Is the expression for $\delta Q / |Q|$ approximate or exact? It looks just too good to be true for $\approx$ CV of somewhat of a messy expression $\prod X_i / \prod Y_i$ without any assumptions. – Just_to_Answer Aug 09 '17 at 01:18
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Variance of a product of two random variables is discussed in the post Variance of product of dependent variables. This is for the general case of dependent scenario and the independent case can be obtained by setting the covariance zero.

Specifically using the result $${\rm Var}(XY)= {\rm cov}(X^{2}, Y^{2}) + [{\rm var}(X) + E(X)^2] \cdot[{\rm var}(Y) + E(Y)^2] - [ {\rm cov}(X,Y) + E(X)E(Y) ]^{2}$$ from the post, you can replace $X$ and $Y$ respectively with $\bar X$ and $\bar Y$. Then taking the square root will get the standard deviation of $\bar X \times \bar Y$.

Then, my suggestion would be to use the expression and consider a "plug-in" estimator. Note that this cannot be computed knowing only the standard deviations, but the computation (with some data manipulation) is possible if the raw data is available.

Just_to_Answer
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