Blog: The Normal Distribution Myth and Statistical Process Control

Introduction

Not all data are normally distributed.  There are many different possible distributions.  It’s a common belief in the world of quality engineering that if data isn’t normally distributed, you can’t apply control charts because normality is a requirement to apply control charts. Turns out, this belief doesn’t hold up – and in this blog we will explain why.

What is the Normal Distribution Curve?

A normal distribution – also known as a bell curve, Gaussian distribution, or Gauss distribution – is a continuous probability distribution that is bell-shaped and symmetric around the mean. It is the most widely used probability distribution in statistics.

In a normal distribution, approximately 68.27% of the data falls within one standard deviation of the mean, 95.45% of the data falls within two standard deviations, and 99.73% of the data falls within three standard deviations.

Normal Distribution in Statistical Process Control

The usage of the normal distribution has roots going back nearly a century. When Statistical Process Control was first introduced in the 1920s by Walter Shewhart, many of the tools we still use today, like X̄ and R charts, were built on assumptions tied to the normal (Gaussian) distribution.

A statistic often used in quality control is the average of multiple samples. A known concept in probability theory is the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the averages approaches a normal distribution as the size of the sample increases, regardless of the shape of the original population distribution. The central limit theorem was first established in a specific case by Abraham de Moivre in 1733, and then generalized and proven in a more general form by Pierre-Simon Laplace in 1810, who is often credited with its overall discovery and demonstration of its importance in probability theory. 

Shewhart used this theorem and established that placing control limits at three standard deviations from the process mean — in both directions — offered a practical balance between the risk of reacting to a false alarm and the risk of not reacting to a true signal. (Economic Control of Quality of Manufactured Product, 1931)

SPC uses Control Charts to monitor data points and identify when a process deviates from its expected normal behavior, indicating potential issues. When a statistic does follow a normal distribution, about 99.7% of data points are expected to fall within three standard deviations of the mean. When a measurement is found outside the 3 sigma limits we assume there is a special cause of variation. That means there’s only a 0.3% chance (or 1 in 370) of a measurement value beyond 3 standard deviations and there is not a special cause of variation.

Although the central limit theorem is applicable, it is not a requirement for using control charts. There are areas in statistics where normality is required. For example in applying Hypothesis Testing, specific tests require normality. For example the F- and t-test require data to be normally distributed.

Also, for capability reporting 99.7 % of the variation is included in the calculation. If data is not normally distributed slight deviations can occur if you assume the data is normally distributed. In case of heavy tails in the distribution it is advised to report the non-normal capability calculation.

Somehow over the years this normality requirement in Hypothesis Testing and Capability resulted in the misconception that normality is required for applying SPC and the use of Control Charts.

So why is normality not required?