tower
  Southwestern
Probability & Statistics  
MATH 241  
   Adventist University
 
Lawrence E. Turner, Ph.D.
 
 
Ye shall keep my sabbaths, and reverence my sanctuary: I am the LORD.
Leviticus 26:2  

 
 

Normal Distribution

extreme probabilities

A statistician is a person who cannot refrain from changing "you're one in a million" to "you're a 4.75 sigma!"


Some companies have adopted a Six Sigma strategy. This was originally developed by Motorola in 1986.

This strategy is normally expressed as an institution-wide philosophy or methodology whereby 99.99966% of the product are free of defects or only 3.4 defects per million (or 1 defect in 294,000). For other types of institutions it may mean that the error rate is less than 3.4 per million (eg. for every million leg amputations, no more than 3.4 will be the wrong leg).

And, clearly some errors are more egregious than others (a wrong letter—a tipo—in a handout is less problematic than entering the wrong final letter grade). The six sigma generally refers to "major" errors or defects.

As can be seen in the table below this value actually comes from Z = 4.5—"six sigma" sounds better! In reality, the justification for the use of 4.5 is that processes do not generally perform as well in the long term as they do in the short term. Therefore, the aim is to acheive a "six sigma" result in the short term.


 
zright-hand tail areacomplement of
right-hand tail area
 
1.0 0.15865 52539 31457   0.84134 47460 68543  1 in 6  
1.50.06680 72012 68858  0.93319 27987 31142  1 in 15  
2.00.02275 01319 48179  0.97724 98680 51821  1 in 44  
2.50.00620 96653 25776   0.99379 03346 74224  1 in 161  
3.00.00134 98980 31630  0.99865 01019 68370  1 in 741  
3.50.00023 26290 79034  0.99976 73709 20966  1 in 4,300  
4.00.00003 16712 41837  0.99996 83287 58163  1 in 31,600  
4.50.00000 33976 73134  0.99999 66023 26866  1 in 294,000  
5.00.00000 02866 51654  0.99999 97133 48346  1 in 3,500,000  
5.50.00000 00189 89564  0.99999 99810 10436  1 in 52,700,000  
6.00.00000 00009 86588  0.99999 99990 13412  1 in 1,013,000,000  

 
 
© 2007–2016, 2017 by Lawrence Turner