Three Sigma Limits Statistical Calculation With Example

What Is a Three Sigma Limit?

A three sigma limit is a statistical calculation in which the data are within three standard deviations from a mean. Three sigma refers to processes in business applications that operate efficiently and produce items of the highest quality.

Three sigma limits are used to set the upper and lower control limits in statistical quality control charts. Control charts establish limits for a manufacturing or business process that's in a state of statistical control.

Understanding Three Sigma Limits

Control charts are also known as Shewhart charts, named after Walter A. Shewhart, an American physicist, engineer, and statistician (1891–1967). Control charts are based on the theory that a certain amount of variability in output measurements is inherent even in perfectly designed processes.

Control charts determine if there's a controlled or uncontrolled variation in a process. Variations in process quality due to random causes are said to be in control. Out-of-control processes include both random and special causes of variation. Control charts are intended to determine the presence of special causes.

Statisticians and analysts use a metric known as the standard deviation to measure variations, also referred to as sigma. It's a statistical measurement of variability showing how much variation exists from a statistical average.

Consider the normal bell curve which has a normal distribution. The farther to the right or left a data point is recorded, the higher or lower the data is than the mean. Low values indicate that the data points fall close to the mean. High values indicate that the data is widespread and not close to the average.

Example of Calculation

Let’s consider a manufacturing firm that runs a series of 10 tests to determine whether there's a variation in the quality of its products. The data points for the 10 tests are 8.4, 8.5, 9.1, 9.3, 9.4, 9.5, 9.7, 9.7, 9.9, and 9.9.

1. First, calculate the mean of the observed data: (8.4 + 8.5 + 9.1 + 9.3 + 9.4 + 9.5 + 9.7 + 9.7 + 9.9 + 9.9) / 10, which equals 93.4 / 10 = 9.34.
2. Second, calculate the variance of the set: Variance is the spread between data points and is calculated as the sum of the squares of the difference between each data point and the mean divided by the number of observations. The first difference square will be calculated as (8.4 - 9.34)² = 0.8836, the second square of difference will be (8.5 - 9.34)² = 0.7056, the third square can be calculated as (9.1 - 9.34)² = 0.0576, and so on. The sum of the squares of all 10 data points is 2.564. The variance is therefore 2.564 / 10 = 0.2564.
3. Third, calculate the standard deviation: This is simply the square root of the variance. The standard deviation = √0.2564 = 0.5064.
4. Fourth, calculate three sigma: This is three standard deviations above the mean. It's (3 x 0.5064) + 9.34 = 10.9 in numerical format. None of the data is at such a high point so the manufacturing testing process has not yet reached three sigma quality levels.
   

The Bottom Line

The term "three sigma" points to three standard deviations. Shewhart set three standard deviation (3-sigma) limits as a rational and economic guide to minimum economic loss.

Around 99.73% of a controlled process will occur within plus or minus three sigmas so the data from a process ought to approximate a general distribution around the mean and within the predefined limits. Data that lie above the average and beyond the three sigma line on a bell curve represent less than 1% of all data points.