Historical
In the 1970s, a Japanese company took over an American Motorola plant that was producing TVs under the Quasar brand^{1}. Under the new management, a change initiative was put in place, and the factory was able to produce televisions with a defect rate 20 times lower than the previous level and that, with the same operators, technologies and machines. Beyond this, Motorola was challenged by increased growth of Japanese companies.
In the early 1980s, Motorola decided to take the quality problem seriously. Bob GALVIN, his general manager at the time, launched the company in a drastic program known as de Six Sigma.
It will be in 1985 that the engineer Bill Smith, based on the work of the “logic filter” of the professor Mikel Harry^{2}, considered the father of Six Sigma, writes a report on the correlation between customer satisfaction (due to the reliability of the product) and the number of nonconformities at each step of the process. He laid the foundation for Six Sigma in developing the MAIC : Measure, Analyse, Improve, Control, which represents a logical approach based on the tools statistics, the SPC, le PDCA… The results of the program were very concrete and the company won the Malcolm Baldrige National Quality Award in 1988, the most prestigious quality award in the United States.
It will only be in 1995, when Professor M. Harry deployed Six Sigma at General Electric that the term Define was added to create the DMAIC that we know.
Today, Motorola is recognized as a world leader in quality, and Six Sigma has spread to other companies like General Electric, Bombardier, Allied Signal and Xerox, and now to the entire industrial and service world..
Introduction to Six Sigma
With the Six Sigma approach, quality is not only seen in terms of defects. Quality is taken in more general terms which is the maximization of value. Thus, a defect can generate a delivery delay or a cost higher than expected. It is therefore a general methodology of defect analysis.
The other advantage of Six Sigma is that it transforms the chaotic nature of variations into clear problems of « yes or no » : either the product meets the customer’s requirements or not. Any exit from a process must meet the expectations of the customer and if it is not the case, the output is considered defective: a coffee served with sugar while the customer requested it without is a defect.
The 6 Sigma methodology relies on rigorous statistical analysis of process data to predict variation. It aims to put in place a strategy to identify the equation :
Y = f(X)
 Y : the average output values of a process
 f(x) : the different influential factors Y
To these two variables are added 3 other parameters, derived from customer expectations :
 Target specifications: These are the product specifications required by the customer and designed by the company.
 Upper Specifications Limit (USL) : the upper limit of tolerance.
 Lower Specifications Limit (LSL) : the lower limit of tolerance.
These two limits determine whether a product is considered defective or acceptable. Any product whose measurements are in the LSL / USL range is considered acceptable even if these measures are different from the target specifications. Any product whose measurements are outside the LSL / USL range is considered defective.
From mastering input factors, the 6 Sigma methodology aims to focus the process on the target while reducing the variability around this target.
The 4 sources of variability are^{2} :
 A design not enough robust, very sensitive to external disturbances.
 Unstable raw materials and elementary parts.
 insufficient process capability.
 Inadequate process standards.
A customeroriented methodology
The 6 Sigma is a customeroriented methodology. The starting point of the 6 Sigma study is the tree of CTQ, the translation of customer expectations. These CTQ represent the backbone of the methodology to measure and compare the project’s progress.
The 6 Sigma organization: towards a change of culture
6 Sigma was designed as a global organization aiming to put in place a profound culture change in the company. A system of qualification of the bearers of the approach exists and new roles are defined.
Six Sigma Mathematical Basics
Sigma, standard deviation, measures the dispersion of a variable. The Sigma number of a process gives the percentage of products whose measurements are in the LSL / USL tolerance range. The table below ^{3} shows the matches between a given number the Sigma and the rate of defects produced by a process for Normal law.
Sigma 
Part ratio without defects 
Number of defects per million opportunities 
Category 
0 
6,68% 
933 193 
Not competitive 
1 
8,86% 
697 672 

2 
69,15% 
308 537 

3 
93,32% 
66 807 
Middle class 
4 
99,38% 
6 210 

5 
99,98% 
233 

6 
99,99966% 
3,4 
Worl class 
To calculate the Sigma level, we rely on the number of defects per million opportunities the DPMOs. The level of Sigma is calculated with the formula :
Sigma level = 1,5 + NORM.S.INV (DPMO/1000000)
With :
DPMO : 1 000 000 * Number of defects in the sample / (Number of defects opportunity * Number of samples
Example of Sigma level
Data We considered for each case 1 opportunity of default.  4 Sigma  6 Sigma 

For a 4 Sigma process, only 4 times the standard deviation of our data falls within our tolerances, which is 99.38% of our data. The rest being defects. For a 6 Sigma process, 6 times the standard deviation of our data falls within our tolerance limits, ie 99.99966% of our data. 

120 million letters sent daily by post in 1015.  750,000 lost per day  408 lost a day. 
80,000 flights a day  500 failed landing per days  1 failed landing every 4 days 
20 consultations per day per doctor per 100,000 doctor  12,000 erroneous prescriptions per day  68 erroneous prescriptions per day 
8,760 hours per year of use of the computer system  55 hours of unavailability per year  1mn 48secs of unavailability per year 
Example of Calculating the Sigma Number
Suppose the accounting department bills customers once the products have been delivered and received by the customers in question. Each time an invoice has been sent, the time required for its preparation and dispatch is recorded.
The following table shows a sample of 30 invoices and the corresponding preparation times.
Time required to prepare and send the invoice for each client
Customer 
Time 
Customer 
Time 
Customer 
Time 
1 
21,5 
11 
20 
21 
16 
2 
10,5 
12 
8,5 
22 
12,5 
3 
15 
13 
15 
23 
15 
4 
12,5 
14 
16 
24 
14 
5 
17,5 
15 
13,5 
25 
18,5 
6 
12 
16 
15 
26 
16 
7 
15 
17 
21,5 
27 
13,5 
8 
16,5 
18 
14,5 
28 
12 
9 
16 
19 
9 
29 
15,5 
10 
13,5 
20 
15 
30 
19 
Le calcul de la moyenne et de l’écart type de cet échantillon donne :
 La moyenne : 15 jours.
 L’écart type : 3,14 days.
Suppose, now, that customers demand to receive their bills within 20 days or less. If the billing process operated at a quality level of 6 Sigma, there would be an average of 3.4 invoices processed with a delay of more than 20 days per million invoices. However, already with a sample of thirty invoices, the number of defects is equal to two (customers 1 and 17). Our process is not 6 Sigma.
Using the DPMO calculation, we get :
DPMO = 1 000 000 * 2 / (1 * 30) = 66 667
Let’s have a Sigma level of 3
Source
1 – F. Voehl, H. J. Harrington, C. Mignosa, R. Charron (2014) – The Lean Six Sigma Black Belt Handbook
2 – M. J. Harry (1985) – Practical experiment design
3 – M. Pillet (2013) – Six Sigma : Comment l’appliquer
M. Pillet (2001) – Appliquer la maîtrise statistique des procédés
L. Prud’Homme (2009) – Performance des comités exécutifs : jeux des affinités et du hasard
S. Den Boer (2006) – Six Sigma for IT management : a pocket guide
L. C. Tang, T. N. Goh, H. S. Yam, T. Yoap (2006) – Six Sigma : advanced tools for black belts and master black belt
R. D. Snee, R. W. Hoerl (2003) – Leading Six Sigma : A step by step guide based on experience with GE and other six sigma companies