If x1,x2,x3...xn is a random sample without replacement from a finite population of Y1,Y2,Y3,.....YN of N values, with population mean** M** and standard deviation** sigma** or variance **sigma^2**,** **then the random sample x1,x2,x3....xn has the sample mean = (sum of all n xi 's)/n = **xbar**, say. Sorry for notational gimmicks done!

The distribution of sample mean, **xbar ** can be proved to have the mean ** M **and the variance = **{sigma^2/n}{N-n)/(n-1)}.**

Therefore,

**variance of the sample mean = variance (xbar) = (sigma^2/n)*{(N-n)/(N-1)} (1)**

**The standard deviation of sample mean= (sigma/n)sqrt{(N-n)/(N-1)} (2)**

Thus the variance of the sample mean of sample of size** n , **drawn from a finite population of** size N **and the population variance, **sigma ^**2 are connected by the relation at (1) and the standard deviation by (2)

The factor **sqrt{(N-n)/(N-1)**} involving population size,**N** and sample size,**n** is called the **finite population correction factor, **or in a abbreviation**, FPCF. **

For n=1, the sample is equivalent to without replacement an this is equivalent to the entire popilation itself. fpc=1 and variance (xbar)=sigma^2.

For large N, the FPCF = (N-n)/(N-1) is nearly 1. But when n is comparatively greater than 5% of the population size, N, we use FPCF to detrmine the variance of the sample mean.

The distribution of sample mean helps us to make the probabilistic statemnts about the sample mean in the theory of Inference. It also makes us determine the confidence limits of the domain of the population parameters like population mean. It also helps us to decide whether our sample is drawn from a population of with particular mean and variance.

For accademic interest you can refer to any text books on Exact Samplig theory and topics like distribution of sample mean and how sample mean tends to Normal Distribution despite parent population not following the Normal Distribution under the influence of Central Limit Theorem.

The central limit theorem and the standard errors of the mean and of the proportion are based on the premise that the samples selected are chosen with replacement. However, in virtually all survey research, sampling is conducted without replacement from populations that are of a finite size *N*. In these cases, particularly when the sample size *n *is not small in comparison with the population size *N *(i.e., more than 5% of the population is sampled) so that *n*/*N *> 0.05, a **finite population correction factor (fpc) or finite population multiplier is** used to define both the standard error of the mean and the standard error of the proportion. The finite population correction factor is expressed as

fpc=sqrt(N-n/N-1)

Where N = Population Size and n = Sample Size

Standard error of the mean for finite populations would be = δ/√n (FPC) and the standard error of the proportion for finite populations = √(p(1-p)/n * FPC.

The effect of the FPC is that the error becomes zero when the sample size *n* is equal to the population size *N*.