Game Development Reference
In-Depth Information
Figure 15-4. The effect of standard deviation on the Gaussian distribution curve
For the Gaussian and any other normal distributions, the standard deviation also indicates
how close a value is likely to be to the mean value of the curve. If values are selected randomly
according to the inverse cumulative distribution function, 68% of them will lie within a distance
of one standard deviation from the mean value. Looking at the
= 0.5 curve from Figure 15-4,
68% of randomly selected values will be between x = 9.5 and x = 10.5.
The distance relations extend further out as well. About 95% of randomly selected values
will lie within two standard deviations from the mean and 99.7% will lie within three standard
deviations. The 68-95-99.7 rule only applies to normal probability functions.
To obtain the cumulative and inverse cumulative distribution functions for the Gaussian
distribution requires the integration of Equation (15.2). Unfortunately, there isn't a closed-form
solution to that problem. Several approximate relations have been developed over the years to
compute the inverse cumulative distribution function for the Gaussian distribution. Abramowitz
and Stegun 1 presented a relation that expresses the inverse cumulative distribution function as
the ratio of two algebraic equations.
σ
2
2.515517
+
0.802853
t
+
0.010328
t
x
=− +
t
(15.3)
2
3
1
+
1.432788
t
+
0.189269
t
+
0.001308
t
The quantity t is a function of the probability, p , which has a value between 0 and 1.
1
t
=
ln
(15.4)
⎝⎠
2
Equations (15.3) and (15.4) were used to generate the curve shown in Figure 15-3.
Equation (15.3) assumes that the mean value is 0 and that the standard deviation is 1.
 
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