Gaussian and Error Functions

An apocryphal story is told of a math major showing a psychology student the formula for the infamous Gaussian or bell-shaped curve, which purports to represent the distribution of human intelligence and such. The formula for 

With the appropriate choice of variables, this gives the normalization condition for the Gaussian function 

The standard deviation, a, commonly called sigma, parametrizes the half-width of the distribution. It is defined as the root mean square of the distribution. The mean square is given by 

To evaluate the integrals (6.85) and (6.86) for the Gaussian distribution, we need the additional integrals 

Since the integrand in the first integral is an odd function, contributions from X 0 and x 0 exactly cancel to give zero. The second integral can be found by taking dida on both sides of Eq. (6.83), the same trick we used in Section 6.9. For the IQ distribution shown in Fig. 6.8, the average IQ is 100 and sigma is approximately equal to 15 or 16 IQ points. 

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