The normal distribution function Φ(z) =
√^1
2 π∫z−∞e−ξ(^2) / 2
dξ and the error function^1
which is defined
erf(z) = √^2
π∫z0e−u^2 du (11 .60)can be related by writing
Φ(z) =
√^1
2 π∫ 0−∞e−ξ(^2) / 2
dξ +
√^1
2 π
∫z
0
e−ξ
(^2) / 2
dξ =
1
2 +
√^1
2 π
∫z
0
e−ξ
(^2) / 2
dξ
and then making the substitutions u=ξ/
√2 ,du =dξ/
√2 to obtain
Φ(z) =^1
2+√^1
2 π∫z/√ 20e−u2 √
2 du =^1
2+^1
2erf
(
√z
2)
(11 .61)The normal probability functions given by equations (11.49) and (11.57) are
known by other names such as Gaussian distribution, normal curve, bell shaped
curve, etc. The normal distribution occurs in the study of various types of errors
such as measurements in the quality and precision control of tools and equipment.
The normal distribution arises in many different applied areas of the physical and
social sciences because of the central limit theorem. The central limit theorem,
sometimes called the law of large numbers, involves consequences of taking large
samples from any kind of distribution and can be described as follows. Perform
an experiment and select nindependent random variables Xfrom some population.
If x 1 , x 2 ,... , x n represents the set of nindependent random variables selected, then
the mean m 1 of this sample can be constructed. Perform the experiment again
and calculate the mean m 2 of the second set of n random independent variables.
Continue doing this same experiment a large number of times and collect all the mean
values from each experiment. This gives a set of average values S={m 1 , m 2 ,... , m N}
created from performing the experiment N times. The central limit theorem says
that the distribution of the set of average values Sapproaches a normal probability
distribution with mean μsand variance σs^2 given by
μs=Mean of the set of averages from a large number of samples =μ
σ^2 s=Variance of set of averages from large number of samples =σ
2
nwhere μand σ^2 represent the true mean and true variance of the population being
sampled. The central limit theorem always holds and does not depend upon the
(^1) There are alternative definitions of the error function due to scaling.