What Is a Three Sigma Limit?
Let me explain what a three sigma limit is—it's a statistical calculation where your data stays within three standard deviations from the mean. According to the empirical rule, that covers 99.7% of the data. In business, three sigma points to processes that run efficiently and deliver high-quality outputs. You use three sigma limits to define the upper and lower control limits in statistical quality control charts. These charts set boundaries for a manufacturing or business process that's under statistical control.
Key Takeaways
- The three sigma limit is statistical data within three standard deviations from a mean.
- Three sigma limits set the upper and lower control limits in statistical quality control charts.
- Data above the average and beyond the three sigma line on a bell curve represent less than 1% of all data points.
- Sigma is a statistical variability measurement showing how much variation exists from a statistical average.
Understanding Three Sigma Limits
You might know control charts as Shewhart charts, named after Walter A. Shewhart, the American physicist, engineer, and statistician who lived from 1891 to 1967. These charts rely on the idea that even perfectly designed processes have some inherent variability in their output measurements.
Control charts help you figure out if there's controlled or uncontrolled variation in a process. Variations from random causes mean the process is in control. But out-of-control processes involve both random and special causes of variation. The point of control charts is to spot those special causes.
As a statistician or analyst, you use standard deviation—also called sigma—to measure variations. It's a straightforward statistical measure of how much variation there is from the average.
Think about the normal bell curve with its normal distribution. The farther a data point is to the right or left, the higher or lower it is compared to the mean. Low values show data points close to the mean, while high values mean the data is spread out and not near the average.
Calculating Three Sigma
Let's walk through an example with a manufacturing firm running 10 tests to check product quality variation. The data points are 8.4, 8.5, 9.1, 9.3, 9.4, 9.5, 9.7, 9.7, 9.9, and 9.9.
First, calculate the mean: (8.4 + 8.5 + 9.1 + 9.3 + 9.4 + 9.5 + 9.7 + 9.7 + 9.9 + 9.9) ÷ 10 equals 93.4 ÷ 10 = 9.34.
Next, find the variance, which is the spread between data points. It's the sum of the squares of the differences from the mean, divided by the number of observations. For instance, (8.4 - 9.34)^2 = 0.8836, (8.5 - 9.34)^2 = 0.7056, and so on. The sum of all squares is 2.564, so variance is 2.564 / 10 = 0.2564.
Then, the standard deviation is the square root of the variance: √0.2564 = 0.5064.
Finally, three sigma is three standard deviations above the mean: (3 x 0.5064) + 9.34 = 10.9. None of the data hits that high, so this process hasn't reached three sigma quality levels yet.
Important Note
Sigma measures how far your observed data deviates from the mean or average. Investors use standard deviation to gauge expected volatility—keep that in mind.
When to Use Three Sigma
You can apply three sigma in various scenarios, such as setting control limits, identifying and analyzing outliers, distinguishing between normal and unusual variation, quality control and improving accuracy in manufacturing, estimating probabilities and outcomes of events, and detecting anomalies in fields like financial analysis.
Three Sigma vs. Six Sigma
The main difference between three sigma and six sigma limits is their accuracy. Three sigma covers three standard deviations from the mean, capturing 99.7% of the data. Six sigma goes to six standard deviations, covering almost all data at 99.99%, making it more precise.
You see three sigma in manufacturing, quality assurance, and industries where some variance is acceptable. Six sigma is for areas needing high data-driven accuracy to cut costs, like manufacturing, finance, healthcare, and IT.
How Are Three Sigma Limits Used?
Three sigma limits set a range for process parameters at 0.27% control limits. You use them to check if process data is within statistical control by seeing if points are within three standard deviations from the mean. The upper limit is three sigma above the mean, and the lower is three sigma below.
What Is Standard Deviation?
Standard deviation is a statistical measurement that calculates the spread of values from their average. It's the positive square root of the variance and shows the difference between variation and the mean.
What Is a Bell Curve?
A bell curve looks like a symmetrical bell-shaped curve that peaks in the middle. It illustrates normal probability, and many graphs and distributions use it. The curve marks data on one, two, and three standard deviations.
The Bottom Line
Three sigma refers to three standard deviations. Shewhart established three sigma limits as a practical guide to minimize economic loss. About 99.7% of a controlled process falls within plus or minus three sigmas, so your data should distribute normally around the mean within those limits. Data beyond the three sigma line above the average represents less than 1% of points.
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