6.
Process or Product Monitoring and Control
6.5. Tutorials


The Normal distribution model  "Normal" data are data that are drawn (come from) a population that has a normal distribution. This distribution is inarguably the most important and the most frequently used distribution in both the theory and application of statistics. If \(X\) is a normal random variable, then the probability distribution of \(X\) is  
Normal probability distribution  $$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{ \frac{1}{2} \left(\frac{x  \mu}{\sigma} \right)^2 } \,\,\,\,\,\, \infty < x < \infty \, .$$  
Parameters of normal distribution  The parameters of the normal distribution are the mean \(\mu\) and the standard deviation \(\sigma\) (or the variance \(\sigma^2\)). A special notation is employed to indicate that \(X\) is normally distributed with these parameters, namely $$ X \sim N(\mu, \, \sigma) \,\,\,\,\,\, \mbox{or} \,\,\,\,\,\, X \sim N(\mu, \, \sigma^2) \, . $$  
Shape is symmetric and unimodal 
The shape of the normal distribution is symmetric and unimodal. It
is called the bellshaped or Gaussian distribution after its
inventor, Gauss (although De Moivre also deserves credit).
The visual appearance is given below.


Property of probability distributions is that area under curve equals one  A property of a special class of nonnegative functions, called probability distributions, is that the area under the curve equals unity. One finds the area under any portion of the curve by integrating the distribution between the specified limits. The area under the bellshaped curve of the normal distribution can be shown to be equal to 1, and therefore the normal distribution is a probability distribution.  
Interpretation of \(\sigma\) 
There is a simple interpretation of \(\sigma\).
95.45 % of the population fall between \(\mu \pm 2 \sigma\) 99.73 % of the population fall between \(\mu \pm 3 \sigma\) 

The cumulative normal distribution 
The cumulative normal distribution is defined
as the probability that the normal variate is less than or equal to some
value \(v\),
or
$$ P(X \le v) = F(v) = \int_{\infty}^v
\frac{1}{\sigma\sqrt{2\pi}} e^{ \frac{1}{2} \left( \frac{x\mu}{\sigma} \right)^2 } dx \, . $$
Unfortunately this integral cannot be evaluated in closed form and one
has to resort to numerical methods. But even so, tables for all possible
values of \(\mu\) and \(\sigma\)
would be required. A change of variables rescues the situation. We let
$$ z = \frac{x  \mu}{\sigma} \, . $$
Now the evaluation can be made independently of \(\mu\) and \(\sigma\); that is, $$ P(X \le v) = P \left(z \le \frac{v\mu}{\sigma} \right) = \Phi \left( \frac{v\mu}{\sigma} \right) \, , $$ where \(\Phi(.)\) is the cumulative distribution function of the standard normal distribution \((\mu=0, \, \sigma=1)\). $$ \phi(z) = \frac{1}{\sqrt{2 \pi}} e^{\frac{z^2}{2}} $$ 

Tables for the cumulative standard normal distribution 
Tables of the cumulative standard normal distribution are given in
every statistics textbook and in the
handbook. A rich
variety of approximations can be found in the literature on
numerical methods.
For example, if \(\mu = 0\) and \(\sigma=1\) then the area under the curve from \(\mu 1\sigma\) to \(\mu + 1 \sigma\) is the area from 0  1 to 0 + 1, which is 0.6827. Since most standard normal tables give area to the left of the lookup value, they will have for \(z = 1\) an area of 0.8413 and for \(z = 1\) an area of 0.1587. By subtraction we obtain the area between 1 and +1 to be 0.8413  0.1587 = 0.6826. 