# Standard Deviation and Normal Distribution in Six Sigma

Standard deviation is a measure of dispersion, or how spread out a data set is. It is calculated by taking the square root of the variance, which is the average of the squared differences between each data point and the mean.(1)

Standard deviation is a useful measure because it is expressed in the same units as the original data, making it easier to interpret.(2) For example, if the standard deviation of a set of heights is 3 inches, it means that the majority of the data points are within 3 inches of the mean.

A normal distribution is a type of probability distribution that is symmetrical and bell-shaped. It is defined by its mean, which is the center of the distribution, and its standard deviation, which determines the width of the distribution. The standard deviation is also known as the “standard error,” as it represents the amount of error in the mean.

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The normal distribution is important because it is used to model many real-world phenomena, such as IQ scores, height, and weight. It is also the basis for statistical tests such as the t-test and z-test, which are used to compare means and determine the likelihood that a result is due to chance.

The normal distribution is defined by a curve, with the majority of the data points clustered around the mean and fewer data points towards the extremes. The curve is shaped like a bell, hence the name “bell curve.” In a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

One of the key characteristics of the normal distribution is that it is continuous, meaning that there is an infinite number of possible values between any two points on the curve. This is in contrast to a discrete distribution, in which only a finite number of possible values exist.

The normal distribution is also known as the Gaussian distribution, after the mathematician Carl Friedrich Gauss, who developed the theory of least squares, which is used to fit a normal distribution to a set of data.(3) Gauss’s work laid the foundation for the field of statistics and has significantly impacted how we analyze data.

## Applications in Six Sigma

In Six Sigma, normal distribution and standard deviation are used to understand data distribution and identify opportunities for process improvement. Standard deviation is often used to identify opportunities for process improvement by identifying processes that have a high level of variability, which can lead to defects or poor quality outcomes.

By understanding the normal distribution and using standard deviation to measure the dispersion of data, Six Sigma practitioners can identify patterns and trends in data and use this information to improve processes and reduce variability. This can lead to higher quality outcomes and increased efficiency.

These techniques are used by Lean Six Sigma Black Belts while leading process improvement projects. They are required learning under the Six Sigma Body of Knowledge for Black Belt and Master Black Belt practitioners. You can gain your Lean Six Sigma Black Belt certification from the Management and Strategy Institute.

#### References:

#### Rei Takako

BS, Business Management

Chicago, IL