Page 350 - DMTH404_STATISTICS

Statistics

Notes The above definition of random sampling holds in both the situations, i.e., in simple random
sampling with replacement (srswr) and in simple random sampling with out replacement (srswor).

24.1 Distinction between Parameter and Statistic

Let P , P ...... P denote the observations on N units of a population and X , X ...... X be a simple
1 2 N 1 2 n
random sample of size n from it.
A parameter is a measure computed from the observation of the population. For example :
P + P +  + P N
2
1
Population Mean ( ) m = ,
N
1 2
2
P -
s
Population Variance ( ) = å ( i ) m , etc. are parameters.
N
In a similar way, a statistics is a measure computed from the observations of a sample. For
example:
X + X +  + X n
2
1
X
Sample Mean ( ) = ,
n
1 2
2
S
Sample Variance ( ) = å ( X - X ) , etc. are statistic.
i
n
Formally, a parameter is any function of population values while a statistic is a function of
sample values.
Very often, the values of various parameters are unknown and these are estimated by the
corresponding statistic. For example, sample mean X is used as an estimator of population
mean m, sample standard deviation S is used as an estimator of population standard deviation s,
etc. The difference between a statistic and the corresponding parameter is known as sampling
error. For example, the sampling error in estimation of m is X - m. It may be noted that the
sampling error is an error caused by pure chance factors.
When we take a random sample X , X ...... X from a population P , P ...... P , the first sample
1 2 n 1 2 N
observation X could be any one of the N population observations P , P ...... P . We know that
1 1 2 N
1
the probability of selection of any one of the population observation is and therefore, we
N
1
can regard X as a random variable which can take values P , P ...... P each with probability .
1 1 2 N N
1 1 1 1
Further, ( ) = P × 1 + P × 2 +  + P × N = å P = m and
E X
1
i
N N N N
Variance of X = E(X - m) 2
1 1
1 2 2 2 1 2 2
P -
P -
= é (P - m ) (P+ 2 - ) m +  + ( N ) m ù = å ( i ) m = s
1
N ë û N
In a similar way, X , X ...... X are all random variables, each with mean m and variance s . The
2
2 3 n
magnitude of covariance between any two of these variables, say X and X, will depend upon
i j
whether the sampling is with or without replacement.
In the case of sampling with replacement, X , X ...... X would be statistically independent and
1 2 n
the Cov(X , X) = 0 for i  j.
i j
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