Combine Normal Distributions

How do you combine multiple normal distributions with different standard deviations and means, to produce one representative normal distribution?

Thank you!
Depends on how you want to combine them. So you could have supplied a little bit more information (it helps to read this).
I am going to assume you mean the sum of multiple normally distributed random variables. In that case you need to look into the Normal sum distribution.


Ambassador to the humans
Oh ok. Now I see where Outlier was coming from. I'm with The Ecologist on this one though. Now let's just hope they do mean the sum of normally distributed random variables and not the product!


TS Contributor
I guess I assumed you can have normal distributions with less than 30 samples. I'm probably mixing up several concepts. I'd think a distribution with more samples than another one should be weighted differently when combined, like a weighted average. Maybe not.
I dont get this. If you have two distributions with wildly different means, and very tight standard deviations, why would you get another normal distribution? It seems like you would just get a bimodal distirbution, or two normal pdfs sepearated by a gap.


Ambassador to the humans
The OP never made their way back to this thread so I don't think we'll get clarification on what they actually were interested in. For everybody else I think a big problem so far is that it's not sufficiently clear what you're trying to convey.

SCE1000 said:
If you have two distributions with wildly different means, and very tight standard deviations, why would you get another normal distribution?
It depends on what you're doing to combine them. If we're just assuming some mixture distribution then your intuition seems fine. If we're taking the sum of two normal random variables with the given means and variances then the resulting variable will have a normal distribution.

@hlsmith - I'm not sure what you were asking exactly.


Omega Contributor
Just curious what two variables they were thinking of adding together.

Trying to wrap my head around if they would be colinear or not. Or what types of scales or units they may have.

Also trying to picture how two variables might be uncorrelated and how their sums would look normal. Say I have heights and I am adding it to some made up variable where those with centrally located heights had extremes values on the other variable. An example would help me understand how in this scenario the sum would be normal. I get the addition of correlated variables but don't intuitive get the sum of two totally different scales.


Super Moderator
hlsmith: Look at the theory underlying the ratio of a sum of U-Statistics and see how this ratio is asymptotically normally distributed.
I ran across this thread while looking for what I believe was the OP's intended question. Below I provide an example and question with that assumption. In my case, I'm interested in knowing how to combine, say, three independent normally distributed variables. That is, I want to know the resultant additive value of three consecutively-occurring, independent, normal functions at a certain confidence.

Let's say a different normal curve describes each of the following activities, in minutes: how long it takes to drive to the dentist (mean=5, stdev=2), how long I spend there (60, 15), and how long it takes to drive back home, fighting traffic (15, 5).

If I wanted to know with 95% confidence that I had allocated enough time in my schedule to accommodate the total trip, how would I find the value of the joint distribution at +2 standard deviations?

This is a simplified but relevant example of what I'm trying to solve, and I think it's in the spirit of the OP's question. Thanks in advance for any help!