In random sampling from a **normal** population with mean
and stadard
deviation ,
the sample mean
has the normal distribution
with mean
and standard deviation
.

When sampling from a nonnormal population, the distribution of
depends on the particular form of the population distribution that prevails. A
surprising result, known as the **central limit theorem**, states that when
the sample size *n* is large, the distribution of
is
approximately normal, regardless of the shape of the population distribution. In
practice, the normal approximation is usually adequate when *n* is greater than
30.

**Whatever the population, the distribution of **
** is
approximately normal when ***n*** is large**.

In random sampling from an arbitrary population with mean
and standard
deviation ,
when *n* is large, the distribution of
is
approximately normal with mean
and standard deviation
.
Consequently,

is approximately *N*(0,1)