# Three Sigma Limits Statistical Calculation With Example

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*— Investopedia / Julie Bang*

> **Definition**
>
> In statistics, the mean less three standard deviations is referred to as the three sigma limit.

## What Is a Three Sigma Limit?

A three sigma limit is a statistical calculation in which the data are within three [standard deviations](https://www.investopedia.com/terms/s/standarddeviation.asp) from a mean. According to the [empirical rule](https://www.investopedia.com/terms/e/empirical-rule.asp), that's 99.7% of the data. Three sigma refers to business application processes that operate efficiently and produce high-quality items. 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.

### 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

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](https://www.investopedia.com/terms/v/variability.asp) showing how much variation exists from a statistical average.

Consider the normal [bell curve](https://www.investopedia.com/terms/b/bell-curve.asp) 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.

## Calculating Three Sigma

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.  Calculate the [mean](https://www.investopedia.com/terms/m/mean.asp) 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.  Determine the [variance](https://www.investopedia.com/terms/v/variance.asp) of the set. Variance is the spread between data points and is calculated as the [sum of the squares](https://www.investopedia.com/terms/s/sum-of-squares.asp) 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) = 0.8836, the second square of difference will be (8.5 - 9.34) = 0.7056, the third square can be calculated as (9.1 - 9.34) = 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.  Calculate the standard deviation. This is simply the square root of the variance. The standard deviation = √0.2564 = 0.5064.
4.  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.  
    

> **Important**
>
> Sigma measures how far observed data deviates from the mean or average. Investors use standard deviation to gauge expected volatility.

## When to Use Three Sigma

Three sigma can be useful when applied to different scenarios, including:

*   Setting control limits
*   Identifying and analyzing outliers
*   Distinguishing between normal and unusual variation
*   [Quality control](https://www.investopedia.com/terms/q/quality-control.asp) and improving accuracy, especially in the manufacturing sector
*   Estimating the probability and the outcome of certain events
*   Detecting anomalies or unusual events in different fields, including financial analysis

## Three Sigma vs. Six Sigma

The primary difference between the three sigma limit and six sigma limit is their degree of accuracy. Remember that the three sigma limit is three standard deviations from the mean. This means that 99.7% of the data falls three standard deviations from the mean. In six sigma, almost all of the data (99.99%) falls within six standard deviations of the mean. This means that the Six Sigma limit is generally considered more accurate.

Three sigma is commonly used in manufacturing, quality assurance, and other industries where some degree of variance is tolerable. six sigma, though, is required in areas where data-driven accuracy is required, especially to cut costs. This includes fields like manufacturing, [finance](https://www.investopedia.com/terms/f/finance.asp), [healthcare](https://www.investopedia.com/terms/h/health_care_sector.asp), and information technology.

## How Are Three Sigma Limits Used?

Three sigma limits set a range for the process parameter at 0.27% control limits. Three sigma control limits are used to check data from a process and to determine if it's within statistical control by checking if data points are within three standard deviations from the mean. The upper control limit is set three sigma levels above the mean and the lower control limit is set at three sigma levels below the mean.

## What Is Standard Deviation?

Standard deviation is a statistical measurement. It calculates the spread of a set of values against their average. It's the positive square root of the variance and defines the difference between the variation and the mean.

## What Is a Bell Curve?

A bell curve gets its name from its appearance: a symmetrical bell-shaped curve that rises in the middle. It illustrates normal probability and several graphs and distributions use it. The single line measures data on one, two, and three standard deviations.

## 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 99.7% of a controlled process occurs 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.

> **Article Sources**
>
> 1.  National Library of Medicine. "[Walter A. Shewhart, 1924, and the Hawthorne Factory](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2464836/)."
>     
> 2.  Six Sigma. "[Six Sigma Principles. Process Capability Index (Cpk) for Business Success](https://www.6sigma.us/process-improvement/process-capability-index-cpk/)."
>     
> 3.  Dataforth. "[an119: Six Sigma: What? Why? How?](https://www.dataforth.com/six-sigma-signal-conditioning)"
>     
> 4.  FasterCapital. "[Defect Rate: Reducing Defect Rates Using Three Sigma Limits.](https://fastercapital.com/content/Defect-rate--Reducing-Defect-Rates-Using-Three-Sigma-Limits.html)"
>     
> 5.  Britannica. "[Standard Deviation](https://www.britannica.com/topic/standard-deviation-statistics)."
>     
> 6.  University of North Dakota Libraries. "[Bell-shaped curve](https://libguides.und.edu/c.php?g=1333878&p=10080475)."