By Issa Bass
 

Six Sigma is defined as a methodology that aims at a quasi perfect production process. Some authors define it as a methodology that aims at a rate of 3.4 Defect Per Million Opportunities, but the 3.4 DPMO remains very controversial among the Six Sigma practitioners.

Why 6? Why σ? And why 3.4DPMO?

To answer these questions, we need to get acquainted with at least three statistical tools: the mean, the standard deviation and the Normal Distribution theory.

In the design phase of their manufacturing processes, businesses correctly identify their customers' needs and expectations. They design products which can be consistently and economically manufactured to meet those expectations.

Every product exhibits particular characteristics some of which are critical to quality (CTQ), because their absence or their lack of conformance to the customers' requirement can have a negative impact on the reliability of the product and on its value.

Because of the importance of the CTQ characteristics, after deciding what to produce, the design engineers set the nominal values and the design parameters of the products. They decide on what would be the best design under current circumstances.

For the sake of our discussion, let's consider a rivet manufacturer. Rivets are pins that are used to connect mating parts. Each rivet is manufactured for a given size of a hole, so it must exhibit certain characteristics such as length and diameter to properly fit the holes it is intended for and correctly connect the mating parts. If the diameter of the shaft is too big, it will not fit the hole and if it is too small, the connection will be too loose to connect the parts.

To simplify our argument, we will only consider one CTQ, the length of the rivet which the manufacture sets to exactly 15 inches.

Variability, the source of defects

But a lot of variables come into action when the production process is started and some of them can cause variations to the process over a period of time. Some of those variables are inherent to the production process itself (and referred to as noise factors by Dr. Taguchi) and they are unpredictable sources of variation in the characteristics of the output.

Because the sources of variation can be unpredictable and uncontrollable, when it is acceptable to the customers, businesses specify tolerated limits around the target.

Our rivet manufacturer for instance, would allow +/- 0.002 inches added to the 15 inch rivets it produces, therefore 15 inch becomes the length of the mean of the acceptable rivets.

The mean is just the average it is the sum of all the scores divided by their number.

Since the all the output will not necessarily match the target, it becomes imperative for the manufacturer to be able to measure and control the variations.

The most widely used measurements of variation are the range and the standard deviation. The range is the difference between the highest and the lowest observed data. Since the range does not take into account the data in between, the standard deviation will be privileged in the attempt to measure the level of deviation from the set target.

The standard deviation shows how the data is scattered around the mean which is the target.

The standard deviation s is defined as:

For a sample.

Where ∑ represents “the sum of”, is the mean, n  is the number of rivets observed and n – 1 is the degree of freedom; it is used to derive an unbiased estimator of the population's standard deviation.

If the sample is greater than or equal to 30 or whole population is being studied, there would be no need for a population adjustment and the Greek letter σ will be used instead of s.

Therefore the standard deviation becomes

Where µ is the arithmetic mean, is rivet observed, N  represents the population observed.

Let's suppose that the standard deviation in the case of our rivet manufacturer is .002 and +/-3σ from the mean are allowed. In that case, the specified limits around a rivet would be 15 +/- .006 inches (.002 x 3 = .006). 

So any rivet that measures between 14.994 inches (15 - .006) and 15.006 inches (15 + .006) would be accepted and anything outside that interval would be considered as a defect.

Evaluation of the process performance

Once the specified limits are determined, the manufacturer will want to measure the process performance, to know how the output compares to the specified limits.

He will therefore be interested in two aspects of the process, the process capabilities and the process stability. The process capability refers to the ability of the process to generate products that are within the specified limits and the process stability refers to the manufacturer's ability to predict the process performance based on past experience.

In most cases, the Statistical Process Control (SPC) is used for that purpose and control charts are utilized to interpret the production patterns.

Since it would be costly to physically inspect every rivet that comes off the line, a sample will be taken and audited at specified intervals of time and an estimation will be derived for the whole production to determine the number of defects.

Normal distribution and process capability

In general, manufactured products are normally distributed and when they are not. The Central Limit Theorem usually applies. So the Normal Distribution is used when a sample is taken from the production line and the probability for rivet being defective is being estimated.

The density function of the normal distribution is:

The curve associated with that function is a bell shaped curve that spreads from -∞ to +∞ and never touches the horizontal line. The area between the curve and the line represents the probabilities for an event to occur and the whole area is estimated to be equal to 1.

In the graph bellow, the area between the USL and the LSL represents the products in conformance and the darkened areas at the tails of the curve represent the defective ones

If the manufacturer uses the sigma scale and sets the specifications to +/-3σ, how many rivets should we expect to be within specification?

Since the area under the Normal curve that uses σ scale has already been statistically estimated, (see table 1 ), we can derive an estimation of the quantity of the products which are in conformance.

Table 1

The probability for a rivet to be between µ - 3σ and µ + 3σ is .9973 and the probability for it to be outside the limits will be .0027 (1 - .9973), in other words 97.73% of the rivets will be within specified limit or 2700 out of a million will be defective.

Let's suppose that the manufacturer improves the production process and reduces the variation to where the standard deviation is cut in half and it becomes .001. We need to bear in mind that a higher standard deviation implies a higher level of variation and that the further the specified limits are from the target µ, the more variation is tolerated and therefore the more poor quality products are tolerated (a 15.0001 inch long rivet is closer to the target than a 15.005 inch long).

The table below shows the level of quality associated to σ and the specified limits (µ + zσ). We can clearly see that the quality level at +/-6σ after improvement is the same as the one at +/-3σ when σ was .002 (14.994, 15.006), but the quantity of conforming products has risen to 99.9999998% and the defects per million have dropped to .002. An improvement of the process has lead to a reduction of the defects. 

Table 2

What does 3.4 DPMO have to do with all that?

We have seen in table 1 that 6σ level corresponds to .002 defects per million opportunities. In fact 3.4dpmo is obtained at +/-4.5σ.

But this only apply a static process in other words to a short term process.

According to the Motorola Six Sigma advocates, small shifts that are greater than  will be detected and corrective actions taken, but shifts smaller than  can go unnoticed over a period of time and will not change.  in the long run, an accumulation of shifts in the process average will lead to a drift in standard deviations of the process.

So in the worst case, the noise factors will cause a process average shift that will result in it being  away from the target, therefore only  will be the distance between the new average process and the closest specified limit.  And a corresponds to 3.4DPMO.

Let's note that manufacturers seldom aim at 3.4 DPMO, Their main objective is to use Six Sigma for the sake of minimizing defects to the lowest possible rates and increase customer satisfaction.

About the author
Issa Bass is the managing editor of SixSigmaFirst. He can be reached at issa@sixsigmafirst.com

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