Example: Say I have a normal distribution that contains responses from both male and female participants with an average score of 50 ± 2.

Lets then say I actually receive this information in the following way; Men have an average score of x = 49 ± 3. Women have an average score of y = 51 ± 4. I can use this to calculate the combined mean of these values z = 50 ± 5 (by summing the errors in quadrature).

Question How much information about the male and female distributions needs to be encoded in order to retain the original error or how do I obtain a more realistic error on Z based on the knowledge that these are essentially part of the same data. Mean, variance and sample size intuitively seem enough to accomplish this.

The trivial case for two normal distributions both with n entries and the same mean would scale like; \begin{align} {\frac{1}{\sqrt{2}}} \end{align}

I suppose I am looking for the general form of this scaling for the error.

  • $\begingroup$ Are you saying you want to compute the standard error of a mean of a mixture distribution from the statistics of the component distributions? $\endgroup$ Commented Jan 19, 2017 at 13:56
  • $\begingroup$ That sounds about right, yes. $\endgroup$ Commented Jan 19, 2017 at 14:14
  • $\begingroup$ Do you know the ns? $\endgroup$ Commented Jan 19, 2017 at 14:17
  • $\begingroup$ Not sure what you're refering to there. Ns being sample size? Ifso, then yes $\endgroup$ Commented Jan 19, 2017 at 14:18
  • $\begingroup$ Yes. In your case, how many males & how many females. $\endgroup$ Commented Jan 19, 2017 at 14:19

1 Answer 1


I found the answer to this question here; How to 'sum' a standard deviation?

I've simulated it robustly and it works out fine. One can take the variance of the mixed distribution as the average of the two component distributions and work out standard error from there.


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