The Central Limit Theorem

1
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.

2
The Central Limit Theorem

• The 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)

3
Example

• Consider a population of die throws generated by
  tossing a die infinitely many times, with a
  resulting probability distribution given by
• P(X)

?X 3.5
1/6
?X 1.7078
X
1
2
3
4
5
6
4
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.

5
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 gt
  30 ), the distribution of possible X values will
  be approximately normal, with
• ?X ?X and ?X ?X / SQRT(n)