Six Sigma — The Math Behind It
In this post, I will explain what is meant by 6-sigma, the holy grail that almost all companies strive to achieve in their operations and production processes and what is the significance of the “6”. Though the concept of Six-Sigma was not new to me, the math behind it was recently explained by my professor in the Operations Management Class in my MBA. I found it interesting and thought of sharing it with everybody.
What is Six Sigma?
Six Sigma refers to a set of techniques and tools that were developed to measure and reduce the number of defects in any process. It was introduced by American engineer Bill Smith while we was working at Motorola in 1986. Subsequently, Jack Welch, former CEO of GE made it the focal point of his business strategy for GE back in 1995
A standard approach to Six Sigma projects is the DMAIC methodology, whose focus is on understanding and achieving what the customer wants. It is widely believed that the key to profitability is achieving what a customer wants. The DMAIC methodology goes as:
- Define(D) -Define the problem, requirements and identify the characteristics that the customer feels have the most impact on quality
- Measure(M) — Identify the variables to be measured and collect relevant data
- Analyze (A) — Develop a set of tools for analysis and identify possible sources of variation and defects
- Improve (I) — Identifying and implementing alternatives for improvement and removing the causes of defects
- Control © — Develop a control plan and establish revised standard measures to maintain performance
A process that is in Six Sigma control is said to produce no more than 3.4 defects per million opportunities (DPMO) i.e no more than 3.4 defects out of 1 million products!!!
Why “Six-Sigma”?
The Greek Letter Sigma is a standard notation in mathematics to denote the standard deviation, which is a measure of how spread out a dataset is.
Suppose there is a manufacturing process that produces a set of 1000 bolts. Due to various error factors (machine error, human error, measurement error etc), all the bolts will not have the same diameter. Assume that the average diameter of the 1000 bolts comes out to be 1.2cm(this is called the process mean). Some bolts will have a diameter less than 1.2cm, some will have diameters above 1.2cm. The question arises? Which bolts do I keep and which ones do I reject?
For this, I need to set specification limits. Suppose I set a specification limit of 0.2cm. This means that all bolts which have diameters ranging from (1.2–0.2) cm to (1.2+0.2) cm i.e 1cm to 1.4cm will be accepted and the remaining will be rejected
(The actual calculations involved behind setting these specification limits constitute a separate field of study known as Statistical Process Control where there are formulae to calculate these specification limits and subsequently the process capabilities. These are beyond the scope of this article)
The term Six-Sigma denotes that if the specification limits are set to 6 standard deviations (sigma) above and below the process mean, the number of defective parts will be around 3.4 out of 1 million parts.
This is how the term “Six Sigma” came about to be. Now, the next question
How Did This 3.4 Come About?
For this, we need to understand the normal distribution, one of the widely used distributions inside and outside of the world of statistics. The Central Limit Theorem states that any process, if run for a sufficiently large amount of time, will eventually reach the normal distribution
If any dataset follows the normal distribution, we can say that:
68.26% of the values lie within 1 standard deviation from the mean
95.44% of the values lie within 2 standard deviations from the mean
99.73% of the values lie within 3 standard deviations from the mean
And so on….
The below table summarizes the results:
Since the area under a normal distribution is 1, Area outside = 1 — Area within. The area outside denotes the % of values that lie outside of the specified region (in the case of manufacturing process, it denotes the % of defects)
The DPMO i.e Defects Per Million Opportunities is obtained by multiplying the % of defects with 1 Million
We see that for a sigma value of 6, the DPMO is 0.001973 and not 3.4 as specified. So, where is the mistake?
Well, there is no mistake here. Experience has shown that most processes do not tend to perform the same way in long-term. Over a period of time, process parameters tend to change, leading to a change in the specification limits. To account for this, a shift of 1.5 sigma is introduced into the calculation. The new values are shown in the table below
We see that the DPMO is 3.39673 which is roughly around 3.4 i.e 3.4 defects per 1 million
An Interesting Trivia To End With!!!
Achieving this six-sigma level of accuracy is unheard of in the corporate world. But, in 1998, Forbes Global Magazine ranked the operational excellence of a certain “company” at a six-sigma level with an accuracy rating of 99.999999%.
Can you identify that “company”?
No? Let me give you a clue
This company consists of “An army of around 5000 people dressed in white dhoti or payjama, white kurta and a white Gandhi topi moving all day long on the roads to assist lakhs of people to enjoy fresh and healthy homemade food”.
Now can you identify who it is?
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Yes. It is the Mumbai Dabbawallas — the army of people who serve lunch daily to Lakhs of people working in Mumbai. You can read more about them here: The Six Sigma Story: Mumbai Dabbawallas
References
Operations and Supply Chain Management by Chase, Shankar, Jacobs
Originally published at http://infinitesimallysmallcom.wordpress.com on April 18, 2021.
