Difference in sample variance formulas So I have a few formulas, and I am not really sure when each formula is applicable. The first formula is: 
$$\newcommand{\Var}{{\rm Var}}
\Var[\bar Y] = \sigma^2_\bar Y = \frac {(N-n)\sigma^2}{(N-1)n}
$$
My understanding, although it may be wrong is that this is the Variance of the population mean?
My next is $\Var[\bar Y_n]\approx (1-f)\frac {\sigma^2}{n}$, where $f=\frac {n}{N}$. My understanding is that this is the sample variance? 
And finally,
$$
s^2_\bar y = (1-f)\frac {s^2}{n}
$$ 
Which I understand to be the variance of the sample mean.
I just want to know when to apply each variance and I have been searching online but I don't really know how to phrase it to get the answers I'm looking for.
 A: Your notes were from a survey sampling class. Survey sampling is a branch of statistics used to deal with FINITE population. Its theories are different from general statistics.
$\sigma^2$ is the variance of $Y$ in the population. (Population mean is constant, and its variance is 0 or it has no variance.)
$\operatorname{Var}[\bar Y]$ is the variance of sample mean. $\operatorname{Var}[\bar Y_n]$ is the same as $\operatorname{Var}[\bar Y]$, the writer did not make them consistent. 
The last one $s_{\bar y}^2$ is the estimate of $\operatorname{Var}[\bar Y_n]$.
A: Your formulas are appropriate when the sample is a non-negligible fraction of a fixed population. 
The variance of $\bar Y_n$ (from a frequentist viewpoint) is the variance over hypothetical replications of drawing a sample of size $n$. If the samples are drawn from the same population of size $N$ there will be some overlap between samples.  On average, each observation in a repeated sample will have a probability $n/N$ of having been in the original sample, so only the $(1-n/N)$ fraction of new observations contribute to the variance.  At the extreme, if $n\approx N$, all the samples will be nearly identical.
If the samples are drawn from a very large population you have $(N-n)/(N-1)\approx 1$ and $1-f\approx 1$.  If the samples are drawn from a data generating process that doesn't correspond to a finite population (eg $Y_i\sim N(0,1)$, $N$ isn't a thing, but the formulas still work if you think of $N$ as infinite, so that $(N-n)/(N-1)=1$ and $f=0$. In practice, even samples from a finite population are often analysed this way: the survey sampling jargon is "sampling with replacement".
