6.5.1. What do we mean by "Normal" data?

6. Process or Product Monitoring and Control
6.5. Tutorials

6.5.1. What do we mean by "Normal" data?

"Normal" data are data that are drawn (come from) a normal distribution. This distribution is arguably the most important and used distribution in both the theory and application of statistics.
If x is a normal random variable, then the probability distribution of x is

The Normal distribution model

The parameters of the normal distribution are the mean m and the standard deviation s (or the variance s²). A special notation is employed to indicate that x is normally distributed with these parameters, namely

x ~ N( m,s) or x ~ N( m,s²).

The shape of the normal distribution is symmetric and unimodal. It is called the bell-shaped or Gaussian distribution , after its inventor, Gauss. (Although De Moivre also deserves credit )

The visual appearance is given below.

A property of a special class of distribution, called probability distributions , is that the area under the curve equals unity. One finds the area under any curve by integrating the distribution (that is, function) and evaluate the result btween the limits. The area under the bell shaped curve of the normal distribution can be shown to be equal to 1, and therefore the normal distribution is a probability distribution.

There is a simple interpretation of s:

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

Unfortunately this integral cannot be evaluated in closed form and one has to resort to numerical methods. But even so, the integral would require tables of all possible values of m ands to yield a meaningful numeric result. To the rescue comes a change of variable

Now the evaluation can be made independently of m and s, that is,

where F(') is the cumulative distribution function of the standard normal distribution (m=0, s=1)

Tables of the cumulative standard normal distribution are given in virtually every textbook dealing with statistics. A rich variety of approximations can be found in literature dealing with numerical methods.

Hence, if m = 0 and s = 1 then the area under the curve for
m - 1s to m + 1s = from 0 - 1 to 0 + 1 = .6827.
Most tables give for z = 1 an area of .8413 and for z = -1 an area of .1587 and by subtraction one gets .8413 - .1587 = .6827.