Three-Sigma Limits: Definition, Applications, and Real-World Examples

Three-sigma limits, often referred to as 3-sigma limits, are a fundamental statistical concept used in quality control and process management. This article delves into the details of three-sigma limits, explaining their significance, applications, and how they influence business processes.

Control charts, also known as Shewhart charts, were developed by Walter A. Shewhart, a prominent American physicist, engineer, and statistician. Control charts are built upon the premise that even well-designed processes inherently exhibit a certain level of variability in their output measurements.

Control charts serve the essential purpose of distinguishing between controlled and uncontrolled variations within a process. Variations resulting from random causes are considered “in-control,” while processes experiencing both random and special causes of variation are deemed “out-of-control.”

To quantify variations, statisticians and analysts employ a metric called the standard deviation, often referred to as sigma. Sigma provides a statistical measure of variability, indicating the extent of deviation from a statistical average.

Imagine a normal distribution represented by a bell curve. Data points located farther to the right or left on this curve signify values higher or lower than the mean, respectively. In simpler terms, low sigma values indicate data points clustering around the mean, while high sigma values suggest data points are more spread out and distant from the average.

Let’s illustrate the calculation of three-sigma limits with a practical example. Consider a manufacturing firm conducting ten tests to assess the quality of its products. The recorded data points for these tests are as follows: 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 = 93.4 / 10 = 9.34.

2. Next, compute the variance of the data set. Variance measures the spread between data points and is calculated by summing the squares of the differences between each data point and the mean, divided by the number of observations. For instance, the first squared difference is (8.4 – 9.34)² = 0.8836, and so on. The sum of these squared differences for all 10 data points is 2.564. Therefore, the variance equals 2.564 / 10 = 0.2564.

3. Calculate the standard deviation, which is the square root of the variance:
√0.2564 ≈ 0.5064.

4. Determine three-sigma, which is three standard deviations above the mean:
(3 x 0.5064) + 9.34 ≈ 10.9.

In this example, none of the data points reach the three-sigma quality level, indicating that the manufacturing testing process has not yet achieved three-sigma quality levels.

The term “three-sigma” refers to three standard deviations. Walter A. Shewhart established three standard deviation (3-sigma) limits as a practical guide to minimize economic loss. These limits set a range for process parameters with a control limit of approximately 0.27%.

Three-sigma control limits are employed to assess data from a process and determine if it is within statistical control. This is accomplished by verifying if data points fall within three standard deviations from the mean. The upper control limit (UCL) is set three standard deviations above the mean, while the lower control limit (LCL) is set at three standard deviations below the mean.

Since approximately 99.73% of a controlled process occurs within plus or minus three sigmas, the data from a process should conform to a general distribution around the mean and within the predefined limits. On a bell curve, data above the average and beyond the three-sigma line account for less than 1% of all data points.

Three-sigma limits are used in statistical quality control to establish upper and lower control limits for processes. They help identify whether a process is within statistical control.

Control charts and the concept of three-sigma limits were developed by Walter A. Shewhart, an American physicist, engineer, and statistician.

Calculating three-sigma limits is crucial for assessing the quality and consistency of a process. It helps organizations identify variations and maintain control over their processes.