:The Central Limit Theorem states that for large sample size :
, even if the population is not normally
distributed.
What this means is that if random samples are drawn from any population
and the sample mean, 
, calculated for each of these samples,
then these sample means would follow the normal distribution with a mean
(or location parameter) equal to the population mean, 
. Thus, the distribution of the statistic,
, would be a normal distribution with
mean 
. The distribution of a statistic is
called the sampling distribution.
, of the sample means would be 
times smaller than the variance of
the population, 
.
This implies that the sampling distribution of the sample means
would have a variance equal to 
(or a scale parameter equal to 
), where 
is the population standard deviation.
The standard deviation of the sampling distribution of an estimator is
called the standard error of the estimator. Thus the standard
error of sample mean 
is 
.
In short, the Central Limit Theorem states that the sampling distribution
of the sample mean is a normal distribution with parameters 
and 
as shown in Figure
3.3.
|
Figure 3.3: Sampling distribution of the sample mean.
The distribution is normal with the mean equal to the population mean
and the variance equal to the |
See Also:
Population Mean, Sample Mean and Variance
Unbiased and Biased Estimators
th fraction of the population variance.