The Central Limit Theorem
Author: John M. Cimbala, Penn State University
Latest revision: 20 August 2007
Introduction
•
It is rare that one can measure something for an entire
population
– instead, a
sample
(or several samples) of the population is measured, and the population
statistics are estimated from the sample.
•
The
Central Limit Theorem
is an extremely useful tool when dealing with
multiple samples.
Multiple samples and the Central Limit Theorem
•
Consider a population of random variable
x
(we assume that variations in
x
are
purely random – in other words, if we would plot a PDF of variable
x
, it would
look Gaussian or normal).
•
The population mean
μ
and the population standard deviation
σ
are not known,
but are instead estimated by taking several samples.
•
We take
N
samples, each of which contains
n
measurements of variable
x
, as
indicated in the sketch to the right.
•
We define the
sample mean
for sample
I
as
1
1
n
I
i
i
x
x
n
=
=
∑
, where index
I
= 1, 2, 3,
…
N
(one for each sample). In other words, we calculate a sample mean in the
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This note was uploaded on 04/05/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Penn State.
 Spring '08
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