Management uses Six Sigma as a technique to maximise the quality of its product. The goal is to achieve the least number of defects per unit of production. A Six Sigma process produces product 99.99966 per cent of the time without errors or defects. This translates to 3.4 defects per million units produced. Lookup tables tabulate the sigma value associated with the number of defects per million units for fast estimates of your systems quality or effectiveness. The value of sigma is a statistical measurement of the standard deviation of a process that has a normal distribution of results. The standard deviation measures the number of results that fall within a defined region of the mean of the distribution. As the number of sigma units increases, the percentage of acceptable results within the defined region of the distribution increases. At Six Sigma, the number of results that are outside the acceptable region of the distribution drops to 3.4.

- Skill level:
- Easy

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### Things you need

- Distribution of events
- Calculator
- Table of erf function values

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## Instructions

- 1
Collect the data on a distribution of events such as a production run. The normal distribution for any event is a bell curve where you would plot results versus frequency of the result. As sigma decreases the number of acceptable events increases.

- 2
Calculate the mean value of the population distribution by summing the value of all the events and dividing by the total number of events. For example, assume the population distribution of 15, 16, 13, 15, 14, 17, 15 and 15. There are eight events in this distribution. The mean value is (15 + 16 + 13 + 15 + 14 + 17 + 15 + 15) / 8 = 120 / 8 = 15.

- 3
Calculate the value of sigma by using the following equation, sigma = sqrt(1/(N -- 1) X SUM(x = 1->N) (event x -- mean)^2 . Using the example given the calculation is, sigma = sqrt (1/7 X [ (15-15)^2 + (16-15)^2 + (13-15)^2 + (14-15)^2 + (17-15)^2 + (15-15)^2 +(15-15)^2] = sqrt( 1/7 X [0 + 1 + 4 + 1 + 4 + 0 + 0] = sqrt( 1/7 X 10) = sqrt(1.43) = 1.2. Therefore, 68.27 percent of the population is within 1.2 of the mean.

- 4
Look up the value of the error function for a normal Gaussian distribution. The error function of n measures the amount of the population under the Gaussian curve between the region of mean+n X sigma and mean-n X sigma, where n is the number of sigma widths being examined. For the values of 1 X sigma, the amount of product within one sigma of the population is 0.683. At Six Sigma, the amount of product within six sigma is 0.999997.

- 5
Multiply the value of the erf function, 0.999997, by 100 to find the percentage. For the example, 100 X erf(6) = 100 X (0.999997) = 99.9997 per cent.

- 6
Couple these values to obtain a full answer. For the example, the distribution has a mean value of 15 with a standard deviation of 1.2 and 99.9997 per cent of the values fall within Six Sigma. The Six Sigma tolerance levels on this population are 15 +/- 7.2.