On the variance of the sample mean from finite population

Journal of Scientific Research

Volume XXXIV No. 2 October 2005

ON THE VARIANCE OF THE SAMPLE MEAN
FROM FINITE POPULATION
Syed Shakir Ali Ghazali1*, Ghausia Masood Gilani**, Muhammad Hussain Tahir***
*Department of Statistics, Govt. S.E. College, Bahawalpur.
**Institute of Statistics, Punjab University, Lahore
***Department of Statistics, Islamia University, Bahawalpur

ABSTRACT
An alternate proof of the variance of the sample mean in case of simple random sampling without replacement (SRSWOR) is obtained. This proof is very simple and avoids the use of expectation.
Key words: Sample mean, simple random sampling, variance, without replacement

sampling.
1. INTRODUCTION
Simple random sampling from a finite population has attracted

much

of

APPROACH I:

the researchers and

Barnett (2002, p.32-35) has given the proof of the

practitioners working in surveys. It is the simplest,

variance of mean as follows:

most preferable and widely used probability

If a sample of size n is drawn from a finite

sampling technique. The variance or standard error

population of size N having y1 , y1 , L , y N units,

of the sampling distribution of mean serves as a

N
n

then there are K =   distinct samples each


basis for efficiency comparison with other sampling methods like stratified random sampling, systematic

−1

N having the same probability   .
n


sampling, cluster sampling etc. The variance of sample mean in case of SRSWOR has been

Let y i , i = 1, 2, L , n be the ith chosen member, then

discussed by Hansen et al. (1953), Murthy (1967),

the probability for obtaining this ordered sequence

Sukhatme and Sukhatme (1970), Cochran (1977),
Jessen (1978), Singh and Chaudhary (1986),

is

DesRaj and Chandhok (1998), Mukhopadhyay

But the probability for obtaining any particular set

(1998), Govindarajulu (1999), Barnett (2002) and

of n distinct population members