There are two (at least!) such approximations.
Once upon a time I wrote a tutorial about a calculation that gives, very nearly, the Annualized Return if you know the Mean Return and Standard Deviation.
In fact, there was a cute geometric picture like so
>And how good is it, this "very nearly" value?
Aah, I'm glad you asked! That's what I want to talk about.
If we let the Mean Return be M and the Variance or (Standard Deviation)2 be V then the Annualized Return, R, is given by:
 R = M - (1/2)V
 (1+R)2 = (1+M)2 - V
>The second one is Figure 1, eh?
Anyway, I tested both on a bunch of stocks (and indexes) over the past ten years and here's what I got:
Well ... uh, that's approximation , as in Figure 1.
>And YOU invented it?
Hardly, but I did generate the formula here
... although I'm sure it's been generated a jillion times before
>And it's just for "annualized" returns, right?
No,  and  are both reasonable approximations not just for annual returns and variance but, for example, monthly ... like so:
>You're talking a monthly compound growth rate?
>Which do you prefer,  or ?
If I'm interested in calculating an Annualized Return (asssuming I only know the Mean and Variance), I'd use .
However, if I want to do some mathematical analysis, I'd use .
>Because it's simpler?
Because it's simpler.
>And you're not concerned about accuracy?
Accuracy? In financial forecasting?
No, I just want to get a feel, to get some indication of how things may vary from one situation to another, to test various scenarios, to ...
>Because it's simpler.
Ummm ... yes.