Normal approximation of a mixture of normal distributions I have noticed that sometimes mixtures of normal distributions can be approximated very well by a simple normal distribution. For instance, the mixture of the normal distributions $\mathcal{N}(0,1)$ and $\mathcal{N}(1,1)$ (with both distributions having a weight $w_i$ of 0.5) is well approximated by $\mathcal{N}(0.5,1.2544)$; the density functions of both distributions differ by about 0.004 or less over the whole continuum. My question is, under which conditions is such a normal approximation of a mixture of normal distributions possible? 
 A: Continued Comment: Single samples of size $n = 1000$ from each of three mixture
distributions illustrate three possibilities:
$U$ is a 50:50 mixture of $\mathsf{Norm}(10, 2)$ and $\mathsf{Norm}(12, 2).$
According to a Shapiro-Wilk test, my sample is consistent with normal: P-value = 0.134.
$V$ is a 50:50 mixture of $\mathsf{Norm}(10, 2)$ and $\mathsf{Norm}(16, 2).$
Judged clearly not normal: $\;\;$ Shapiro-Wilk P-value < 0.0005.
$W$ is a 50:50 mixture of $\mathsf{Norm}(10, 2)$ and $\mathsf{Norm}(13, 6).$
Judged clearly not normal: $\;\;$Shapiro-Wilk P-value < 0.0005.
Different samples from these distributions may give somewhat different
results, but the idea survives: mixtures of distributions with approximately
equal means and equal variances are better approximations to normal.
The graphs below show my three samples along with the density functions
of the three mixture distributions.

Note: Heights of humans are often modeled as approximately normal.
They are close to a 50:50 mixture of women and men, whose average heights
differ by perhaps 3-4 inches, which is smaller than two standard deviations.
Heights might be more closely modeled as normal by looking at women and men separately.
