The Central Limit Theorem

For simple random samples from any population with finite mean and variance, as n becomes increasingly large, the sampling distribution of the sample means is approximately normally distributed.

The Central Limit TheoremThe Central Limit Theorem states that under rather general conditions, sums and means of samples of random measurements drawn from a population tend to possess, approximately, a bell-shaped (normal) distribution in repeated sampling.

Thus, X = X and X = X / SQRT(n)

Example

Consider a population of die throws generated by tossing a die infinitely many times, with a resulting probability distribution given by:

P(X)

X1 2 3 4 5 6

1/6X = 3.5

X = 1.7078

Example

Draw a sample of five ( n = 5 ) measurements from the population by tossing a die five times and record each of the five observations. Calculate the sum of the five measurements as well as the sample mean. For experimental purposes repeat the sampling procedure 100 times or preferably an even larger number of time.

The Central Limit Theorem

If random samples of n observations are taken from any population with mean X and standard deviation of X , and if n is large enough ( n > 30 ), the distribution of possible X values will be approximately normal, with

X = X and X = X / SQRT(n)