Download The Central Limit Theorem

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 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 bellshaped (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)
X = 3.5
1/6
X = 1.7078
1
2 3 4 5
6
X
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)