# Six Sigma Calculation

The basis of Six Sigma calculation is formed by the properties of the normal distribution curve (or the bell curve). Most processes in life follow a normal distribution - and that is good for us. Why? Because we only need to know two things about the normal distribution to get a whole lot of information about it. The two things we need to know are the mean and the standard deviation (sigma). The properties of the bell curve are as such that we can know the probability of getting each value within the curve (probability distribution function) and we can also know the probability of getting a value less than or equal to a number of interest within the curve (cumulative distribution function).

For a process to be at Six Sigma level, it needs to have +/- 6 standard deviations within the specification limits in the short term and +/- 4.5 standard deviations within the limits in the long term. Why 4.5 standard deviations, you may ask. Why not 6 standard deviations throughout? After all, it's called Six Sigma!

From experience, the pioneers of Six Sigma had seen that most processes tend to have approximately a 1.5 sigma shift toward the specification limit. Therefore, for a process to be identified as a Six Sigma process in the long term, it needs to have 4.5 standard deviations within the specification limits. There is quite a lot of debate about this 1.5 sigma shift and not all experts agree on this. However, since the pioneers of the technique have already defined this, we will not argue with their wisdom and we shall avoid confusion! Just keep it simple and remember that a Six Sigma process will have 3.4 defects or less per million parts or transactions in the long term.

For this Six Sigma calculation, we will use the example of jeans manufacturing. If you haven't already, please see our tutorial on the objectives of Six Sigma to get a better understanding of this example.

Let's say a line in your factory is manufacturing 32 inch waist size jeans. You want to know how good your process is. To do the six sigma calculation, you will need to 4 pieces of data:
- the mean or average waist size that is coming off this line,
- the standard deviation (sigma) of the waist size that is coming off this line,
- the lower spec limit (usually specified by the customer),
- and the upper spec limit (usually specified by the customer).

To get the first two pieces of data, you will need to collect samples of jeans coming off the line and measure their waist size. For the last two pieces of data, you will need to check the specifications given by the customer. Let's say the customer specification says the jeans must have a waist size of 32 +/- 1 inches. This means that the customer is willing to accept anything that is between 31 and 33 inches (we know this is lenient, but let's keep things simple for the purpose of this six sigma calculation). Anything outside of that is considered to be a defect.

Now, to get the mean and standard deviation of your process, let's say you went and randomly collected 30 samples of product that came off the line over the last few months. You then measured the waist size for each sample and got the values below:

 Sample Waist Size 1 33.515 2 33.175 3 33.437 4 32.550 5 32.306 6 32.592 7 32.511 8 33.141 9 32.973 10 33.305 11 33.542 12 32.572 13 32.789 14 33.767 15 32.856 16 32.676 17 32.714 18 32.556 19 32.785 20 32.935 21 32.895 22 33.129 23 32.277 24 32.519 25 32.830 26 33.128 27 32.773 28 33.184 29 33.093 30 33.656

You have collected the data, but in order to do the six sigma calculation, you will need to find out the mean and standard deviation from this data. You can input these values into excel and use simple formulas to find the mean and standard deviation of the process. Use the formula =AVERAGE() to get the mean and the formula =STDEV() to get the standard deviation for the list of values. The mean of your current process is 32.939 and the standard deviation is 0.394.

After you have this, you need to subtract your lower spec limit from your mean (AVG - LSL) and also subtract your mean from your upper spec limit (USL - AVG). You then divide the value which gave the lower result by the standard deviation. So, in this example, AVG - LSL is 32.939 - 31. This gives 1.939. USL - AVG is 33 - 32.939. This gives 0.061. Since the latter number is the lower one, we will divide that by the standard deviation. This will be 0.061 / 0.394 which gives 0.155. Voila! There is your sigma level. It's 0.155!

Although you must be happy that you performed the six sigma calculation and found the sigma level, you probably are not very happy with the process. 0.155 is far from the the 4.5 sigma level we need to be considered a Six Sigma process. Another great thing about the normal distribution is that you can determine the probability of getting defects from your process once you know your sigma level. This is done by using the cumulative distribution function. You can use the formula =NORMSDIST() to find this. In the brackets, enter the negative of your process' sigma level. For this example, I would type in "=NORMSDIST(-0.155)" in to Excel. It returns the value 0.4384. That is alarming! This means that my current process has a 43.84% probability of producing a defect!

To improve this process, you will have to use Six Sigma tools and concepts to change the process so that the mean is more centered between the LSL and the USL. You will also want to reduce the variation (by reducing the standard deviation) in the process. Once you've done this, you can then collect some new samples and do the same six sigma calculation. Hopefully, your new sigma level will be much closer to 4.5.

TIP: Type "=NORMSDIST(-4.5)" into Excel and see what you get!

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