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Central Limit Theorem
Example: Parabola

 

 

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Examples:

SEbullet_3.gif (95 bytes) Uniform
SEbullet_3.gif (95 bytes) Triangle
SEbullet_3.gif (95 bytes) Inverse
SEbullet_3.gif (95 bytes) Parabola
SEbullet_3.gif (95 bytes) Summary

The CLT is responsible for this remarkable result:

The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal.

Thus, the Central Limit theorem is the foundation for many statistical procedures, including Quality Control Charts, because the distribution of the phenomenon under study does not have to be Normal because its average will be. (see statistical fine print)

 

Furthermore, this normal distribution will have the same mean as the parent distribution, AND, variance equal to the variance of the parent divided by the sample size.

 

clt_parabola_01.gif (1973 bytes)
Here's an example:

The bi-modal, parabolic, probability density on the left is obviously non-Normal.  Call that the parent distribution.

NonNormal Distribution of x

 

clt_parabola_02.gif (2105 bytes)
To compute an average, Xbar, two samples are drawn, at random, from the parent distribution and averaged. Then another sample of two is drawn and another value of Xbar computed.  This process is repeated, over and over, and averages of two are computed.  The distribution of averages of two is shown on the left.

Distribution of Xbar, when
sample size is 2

 

clt_parabola_03.gif (2114 bytes)
Repeatedly taking three from the parent distribution, and computing the averages, produces the probability density on the left.

Distribution of Xbar, when
sample size is 3

 

clt_parabola_04.gif (2107 bytes)
Repeatedly taking four from the parent distribution, and computing the averages, produces the probability density on the left.

Distribution of Xbar, when
sample size is 4

 

clt_parabola_08.gif (2019 bytes)
Repeatedly taking eight from the parent distribution, and computing the averages, produces the probability density on the left.

Distribution of Xbar, when
sample size is 8

 

clt_parabola_16.gif (2025 bytes)
Repeatedly taking sixteen from the parent distribution, and computing the averages, produces the probability density on the left.

Distribution of Xbar, when
sample size is 16

 

clt_parabola_32.gif (2072 bytes)
Repeatedly taking thirty-two from the parent distribution, and computing the averages, produces the probability density on the left.

Distribution of Xbar, when
sample size is 32

 

Notice that when the sample size approaches a couple dozen, the distribution of the average is very nearly Normal, even though the parent distribution looks anything but Normal. Here is an automation of the change in probability density as the sample size increases:

CLTparabola.gif (9935 bytes)

 

Statistical fine-print: The distribution of an average will tend to be Normal as the sample size increases, regardless of the distribution from which the average is taken except when the moments of the parent distribution do not exist.  All practical distributions in statistical engineering have defined moments, and thus the CLT applies

The Cauchy is an example of a pathological distribution with nonexistent moments.  Thus the mean (the first statistical moment) doesn't exist.   If the mean doesn't exist, then we might expect some difficulties with an estimate of the mean like Xbar.

Further discussion of the statistical fine-print

(return to text)

 

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Last modified: June 08, 2014