Normal Distribution (AKA Gaussian Probability Distribution)

The Normal distribution is used to analyze data when there is an equally likely chance of being above or below the mean for continuous data whose histogram fits a bell curve. Statisticians refer to the normal curve as the Gaussian Probability Distribution, named after Gauss.

Entertainingly, when students ask for a professor to grade on a curve, they probably don’t know that would mean 50% of the students would receive below a 50 or less than a D!

Basic Assumptions:

• Normal Distribution is the most widely known symmetric distribution for continuous data.
• Symmetrical distribution about the mean (a bell-shaped curve)
  • They will never be perfect unless you have an infinite data set.
• Commonly used in inferential statistics.
  • This is often the most used distribution in Six Sigma.
• Family of distributions described by m and s
• The peak of the normal curve indicates the average, which is the center of process variation. An average of a group of numbers indicates the central tendency.
• If there is a normal curve, then there is no undue influence in the process.
• Is symmetric:
  • Many other distributions can be symmetric under the right conditions, including Binomial & Chi Square.
  • Do NOT assume that symmetric data is normally distributed.

When to Use Normal Distribution

• When data is grouped around the mean, there is an equal probability of being above or below the mean (50% above & 50% below the average).
• If we can transform data to behave like a normal distribution, then we do it! Much easier to work with data in this shape.
  • Ex. If we have to take the log of values or subtract a number or perform some other operation on the data, then do it.
• Use when the histogram fits a bell curve.
• Use when the goodness-of-fit statistic is less than the selected P-value (usually 0.05).

Uses Include:

• A Normal Distribution is used to test population means from sample data.
• Use a histogram to determine if data are normally distributed.
• Probabilistic assessments of the distribution of time between independent events taking place at a constant rate.
• The shape can be used to describe constant failure rates as a function of usage.
• The standard normal or t-distributions are most likely used to compare two process means.

Formulas for Standard Normal Distribution

In a normal distribution, 68% of the data will occur within +/- 1 standard deviation.

• e = constant (2.71828) – Poisson constant
• x = control variable – (data being studied)
• µ = population mean
• σ = population standard deviation

Formulas for Population mean, Variance, and Standard Deviation.

N (capital N) refers to Population.

In this case, σ^2 is the variance.

Formulas for Sample mean, Variance, and Standard Deviation.

n (lowercase n) refers to sample size.

In this case, s^2 is the variance.

The population equations differ from the sample equations because we wish to reduce the “degrees of freedom” or increase our confidence in the sample.

Variation and Bell Shape

Adjusting to Center

You do not want to adjust an ongoing process to “center it.” This increases variation. The more you do this, the more the operator unduly influences the process, and the less the distribution will be shaped like a bell. See Quincunx demonstration.

Center of Process Variation

The peak of the normal curve indicates the average, which is the center of process variation.

5Ms & 1 P

When you have a bell-shaped curve, none of the 5 Ms or one P are unduly influencing the process.

Additional Notes:

• Is your process is following a normal distribution?
• How to transform data into a normal distribution.
• Process control for non-normal distributions.

Normal Distribution Videos

Good basic description of normal distribution

What is a standard normal distribution?

ASQ Six Sigma Black Belt Certification Normal Distribution Questions:

Question: For a normal distribution, two standard deviations on each side of the mean would include what percentage of the total population? (Taken from ASQ sample Black Belt exam.)

Answer:

A: 95%. For this question, you need to remember that nearly all of a process’s outputs will be within 6 sigmas – or 6 standard deviations. 2 standard deviations on each side of the mean would be 95% of all outcomes.