Relationship between number of replications and the standard deviation of entry means

Question

I'm working at a plant breeding company. During some spare time I recently did some exploring of our data (yield trials), mostly just out of curiosity and wanting to learn R better.

I started by plotting standard deviation of means created with different number of replications (in the actual data each color would represent a particular seed lot grown in a particular environment). Like this (examples are not actual yield data):

Figure 1. Standard deviation of means created from different number of normally distributed numbers.

That looks perfectly understandable to me. I then got the idea to plot the standard deviation relative to the 1-rep standard deviation:

Figure 2. Standard deviation, relative to standard deviation in 1 rep, of means created from different number of normally distributed numbers.2

Obviously this relative standard deviation has a strong relationship with the number of replications. I have been thinking about it a while, but I'm having trouble to formulate the math behind this. So my question is this:

Can someone please explain to me, in mathematical terms, why we see the relationship visualized in the second figure?

(I have a feeling that the answer will leave me feeling utterly stupid for asking. I also realize that this may sound an awful lot like I'm trying to get you to do a homework assignment for me, but I'm really just curious and have access to loads of data but not a lot of training in statistics and mathematics. We have a statistician that I could have tried to ask, but he's on vacation.)

Answer 1

Your second curve looks close to the line $y=1/\sqrt x$: so near the points $(4,0.5)$, $(16,0.25)$ etc.

This because the standard error of the mean is proportional to the square root of the sample size, and you have removed the constant of proportionality by measuring relative to the standard error with one replication.