# 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.

• where did you get these formulas from? May 26, 2017 at 23:36
• @lucasfariaslf My university notes, why do you ask? May 26, 2017 at 23:39
• Because I can't remember seeing it before, but maybe this could help you situate yourself: talkstats.com/showthread.php/… May 27, 2017 at 0:14
• You will need to provide some context for these. What precedes these in your notes? What are the topics of the sections they show up in? Etc. May 27, 2017 at 1:17

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 $\operatorname{Var}[\bar Y_n]$.

• I'd say there may be a 'problem' if the student doesn't know what class he is taking. May 27, 2017 at 18:45

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".