# What is 'finite population multiplier'?

neela | Student

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.

grgsiocl | Student

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.