Can someone help me to find a study/dataset of height differnces of men and women including normal distributions and their standard deviation/s? i am currently interested in a study or a valid data set of height differences of men and women. What I am explicitly looking for is something with normal distributions and their standard deviation/s. But unfortunately i couldn't find something like that so I thought maybe someone can help me here =)?
 A: Mixture distributions of normal heights are not necessarily normal.
(1) 50:50 mixture: $\mathsf{Norm}(\mu = 65, \sigma=2.5)$
and $\mathsf{Norm}(\mu =69, \sigma=3.5).$ Flat mode and moderately skewed.
curve(.5*dnorm(x,65,2.5)+.5*dnorm(x,69,3), 55, 80, lwd=2, 
      ylab="Density", xlab="Height",
      main="Mixture of Two Normal Distn's: Flat mode")
 abline(h=0, col="green2")
 curve(.5*dnorm(x,65,2.5), add=T, lty="dotted", col="blue")
 curve(.5*dnorm(x,69,3), add=T, lty="dotted", col="brown")


(2) 50:50 mixture: $\mathsf{Norm}(\mu = 63, \sigma=3)$
and $\mathsf{Norm}(\mu =70, \sigma=3).$ Bimodal
curve(.5*dnorm(x,63,3)+.5*dnorm(x,70,3), 55, 80, lwd=2, 
      ylab="Density", xlab="Height",
      main="Mixture of Two Normal Distn's: Bimodal")
 abline(h=0, col="green2")
 curve(.5*dnorm(x,63,3), add=T, lty="dotted", col="blue")
 curve(.5*dnorm(x,70,3), add=T, lty="dotted", col="brown")


Note: "Merging" two populations results in a population with a mixture distribution. By contrast. the sum of two normal random variables is always normal.
