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The
Gaussian Distribution
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The Gaussian Distribution, also
called the Frequency Curve, Bell Curve, or Normal Distribution, is one
of the most widely studied topics in all mathematics. Two of the
most common variations of the equations are . . .
As a probability function:
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As the
so-called Standard Normal Distribution:
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The Deutsche Mark
note with Gauss' picture and his hallmark distribution curve has been
replaced in circulation by the Euro.
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The strength of the Gaussian Distribution is that it is often a very
good approximation. This assumption is based on the Central Limit
Theorem studied in the Calculus. The "CLT" proves that the mean
of any data set, with a distribution having both a finite mean and finite
variance, tends to be Gaussian. This implies that test scores, height,
weight, etc., when graphed will tend to have a "bell" shape, with very few
at either the high or low end.
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From Roger Thatcher's
address to the BSHM Christmas meeting on December 13, 2003:
Re: Thatcher's employment as a statistician in the Ministry of Labour
or Department of Employment.
| "I made some analyses
of the earning statistics and eventually produced two papers for the Royal
Statistical Society. One interesting result concerned the earnings
of manual men. These followed a lognormal distribution. That
is to say, the logarithm of their earnings has a normal distribution, which
is now sometimes called the Bell curve (though earlier it was known as the
Gaussian distribution, originally discovered by de Moivre). There had
been earlier surveys of manual earnings at various dates right back to 1886
and these had been lognormal too. What was extremely unexpected, though,
was that these lognormals all had almost exactly the same variance. This
meant that although the level of earnings had changed enormously since 1886
in terms of £ per week, the percentage differences between higher paid
and lower paid workers had hardly changed at all. I remember that one
day a whizz-kid from Downing Street came round to tell us that there was
going to be a brand new incomes policy, which this time would change the
distribution of earnings. I said: 'Well, it is a lognormal distribution.
What shape do you want it to be?' " |
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References
< http://curvebank.calstatela.edu/famouscurves/famous.htm#FrequencyCurve
>
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Click here for
a view of the spread of a Normal Distribution.
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Weisstein, Eric W.,
CRC Concide Encyclopedia of Mathematics, CRC Press, 1999,
p. 716.
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| For Mathematica®
code that will create many variations of this curve see Gray, A.,
MODERN DIFFERENTIAL GEOMETRY of Curves and
Surfaces with Mathematica®, 2nd. ed., CRC Press,
1998, pp. 393-394.
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