Normal Distribution

The normal distribution is a widely used probability density function to describe data. The distribution is defined by two parameters (Mean) and (STDEV). The curve is also called bell curve due to it shape. This distribution is widely used in statistics because data is often normally distributed.

Influence of the Mean the Red curve has a higher mean	Influence of the STDEV The Red curve has a bigger STDEV

The formula probably density function of the normal distribution is:

Central limit theorem

The sum of a large amount of independent distributions results in a approximately normally distributed data. Even if the independent distribution are not normally distributed. Below are 5 uniform distributions combined and after every extra data-set that is added the result of the Anderson Darling normality calculation is higher (0.00, 0.06, 0.18, 0.29).

Due to the Central limit theorem many data is normally distributed because the data is affected by many different factors.

Data file

Sigma level

A sigma level of 1 is one time the STDEV.

Two sided table
Cpk	Sigma level	% out of tolerance	PPM out of tolerance
0.33	1.0	31.73	317310.508
0.50	1.5	13.36	133614.403
0.67	2.0	4.55	45500.264
0.83	2.5	1.24	12419.331
1.00	3.0	0.27	2699.796
1.17	3.5	0.05	465.258
1.33	4.0	0.01	63.342
1.50	4.5	0.001	6.795
1.67	5.0	0.0001	0.573
1.83	5.5	0.000004	0.038
2.00	6.0	0.0000002	0.002

Example

Sigma level of 2 => 100-(34.1 + 13.6 + 34.1 + 13.6) = 4.6%

Links

• https://en.wikipedia.org/wiki/Normal_distribution
• https://en.wikipedia.org/wiki/Central_limit_theorem
• http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm