# How a Sampling Distribution Is Affected When the Distribution Is Not Normal

In statistics, if a population* X *has any distribution that is *not *normal, or if its distribution is unknown, you can’t automatically say the distribution of the sample means

has a normal distribution. But incredibly, you can use a normal distribution to *approximate* the distribution of

— if the sample size is large enough. This momentous result is due to what statisticians know and love as the Central Limit Theorem.

The *Central Limit Theorem* (abbreviated *CLT*) says that if* X *does* not *have a normal distribution (or its distribution is unknown and hence can’t be deemed to be normal), the shape of the sampling distribution of

is *approximately* normal, as long as the sample size, *n,* is large enough. That is, you get an *approximate* normal distribution for the means of large samples, even if the distribution of the original values (*X*) is* not *normal.

Most statisticians agree that if *n* is at least 30, this approximation will be reasonably close in most cases, although different distribution shapes for *X* have different values of *n* that are needed. The less “bell-shaped” or “normal looking” the distribution of the original values of *X *are, the larger the sample size for the sample means will need to be. The larger the sample size (*n*), the closer the distribution of the sample means will be to a normal distribution.