# How to Calculate 3 Sigma

by Gerald Hanks; Updated September 26, 2017Statisticians use the standard deviation, also known as "sigma," to measure how far a piece of data is from the average. In the classic Bell curve, the further to the right the data lies on the curve, the higher that data is than the statistical average. Business analysts use terms such as "three sigma" to represent processes that operate efficiently and produce the highest-quality products.

## Importance of Three Sigma

While the idea of calculating standard deviations and three sigma may appear to be an esoteric statistical problem best left to textbooks, these calculations have important business applications. On a traditional Bell curve, data that lie above the average and beyond the three sigma line represent less than 1 percent of all data points. According to scientist and statistician Walter Shewart, three sigma represents the boundary between "ordinary" and "extraordinary."

## Calculate the Average

The average -- or statistical mean -- of a data set is calculated by finding the sum of all data points and dividing that sum by the number of data points. The average represents the starting point for finding the standard deviations. For example, a series of eight tests gives data points of 9.0, 9.1, 9.3, 9.4, 9.5, 9.6, 9.8 and 9.9. The average of these data points is (9.0 + 9.1 + 9.3 + 9.4 + 9.5 + 9.6 + 9.8 + 9.9)/8, or 9.45.

## Calculate the Variance

The variance represents the spread between the data points. To find the variance, generate a new data set consisting of the difference between each data point and the mean, then squaring the difference. The variance is the average of these data points. For the data set in the previous section, the first data point to calculate the variance would be (9.45-9.0)^2, or (0.45)^2, or 0.2025. The next data point would be (9.45-9.1)^2, or (0.35)^2, or 0.1225, and so on. The sum of the "difference squares" is 0.7, and the variance is 0.7/8, or 0.0875.

## Calculate Three Sigma

The standard deviation is found by identifying the square root of the variance. In the data set above, the standard deviation is the square root of 0.0875, or 0.2958. Three standard deviations is three times the square root, or 0.8874. Three sigma is three standard deviations above the mean. In the data set above, three sigma is 9.45 + 0.8874, or 10.3374. Since none of the data reach that high, the testing process has not yet reached "three sigma" quality levels.

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