# 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? – Lucas Farias May 26 '17 at 23:36
• @lucasfariaslf My university notes, why do you ask? – user162934 May 26 '17 at 23:39
• Because I can't remember seeing it before, but maybe this could help you situate yourself: talkstats.com/showthread.php/… – Lucas Farias May 27 '17 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. – gung May 27 '17 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. – wolfies May 27 '17 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".