# The Difference Between 3 Sigma & 6 Sigma

by Elliott Taylor; Updated September 26, 2017Sigma, or standard deviation, is a widely used measure of the variability inherent in a population or sample. The difference between three sigma and six sigma is what percentage of the total observations in a data set falls between the mean and the upper limit specified by the particular sigma value.

### What Sigma Means

Sigma is a mathematical term used interchangeably with the phrase "standard deviation." The word "sigma" comes from the Greek symbol used by mathematicians to represent the standard deviation. The standard deviation of a sample is a measure of the average variability between the mean, or average, of a sample or population, and the individual data points that make up the total sample. The standard deviation of a sample has a numeric value.

For instance, if the mean of a process is 100, the standard deviation might be 25. Any values falling between 75 and 125 would be said to fall within plus or minus one standard deviation, or one sigma.

### Calculating The Standard Deviation

The standard deviation of a sample can be calculated as follows: First, calculate the mean of the sample. The mean is calculated by adding all the numbers in a set -- for this example we will use 1, 2, 3, 4, and 5 as our sample points -- and dividing the total by the number of samples. So 1+2+3+4+5 = 15. Divide 15 by the 5 data points and you get a mean of 3.

Next, calculate the variance: This is done by subtracting each sample point from the mean, squaring that number, then taking the average of those squared numbers. So 1-3 = minus-2, which when squared becomes 4, 2-3 = minus-1, which squared becomes 1, and so on. The sum of squares of this sample is 4+1+1+1+1 = 8, and the average, or variance, is 1.6.

Finally, take the square root of the variance to get the standard deviation. The square root of 1.6 is 1.26. So the standard deviation of a sample containing the numbers 1, 2, 3, 4, and 5 is 1.26.

A sample size of at least 30 is recommended for most statistical analyses. The previous example includes only five observations for simplicity's sake.

### The Difference Between Three Sigma and Six Sigma

The difference between three and six sigma is what percentage of the observations that make up the total sample or population will fall between the sigma level and the mean. The rule of "68-95-97" states that approximately 68 percent of the observations in a normally distributed sample fall within plus or minus one standard deviation from the mean, while 95 percent fall within two standard deviations, and just over 97 percent fall within three standard deviations.

In a sample, if the mean is 50 and the standard deviation is 5, minus three sigma would be 35 and plus three sigma would be 65. So if you are controlling your process to within three sigma of the mean, 97.44 percent of the observations in your sample will have values between 35 and 65.

If your process operate at a six sigma level, however, the percentage of points falling between your upper and lower limits increases to 99.99966 percent, meaning that only 3.4 out of every 1,000,000 observations would be outside the expected limits. In the previous example, minus six sigma would be 20 and plus sigma would be 80, increasing the spread between the upper and lower control limits from 30 to 100.

### Tips and Warnings

When a given sigma level is talked about, whether it is one, three, or six sigma, the expression refers to plus or minus the given sigma level. So a sigma level of three would have a spread of six sigma total and a sigma level of six would have a spread of 12 sigma total.

The six sigma curve assumes a normally distributed sample population. In order to be able to assume normal distribution, your sample size must be 30 or more. Just because the variability in a process falls within six sigma does not mean that the process is tightly controlled. If a process has a small standard deviation, six sigma might represent only 10 percent variance from the mean, but if the standard deviation is large, six sigma might represent a control limit 80 percent higher or lower than the mean.

It's very cumbersome to calculate variance and standard deviation by hand. See the Resources section for easier calculation tools.