Normal Distribution

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Distribution - Normal

References: Dr. Mikel J. Harry, Juran's Quality Control Handbook


The Normal Distribution is used to determine the probability of an event regarding a population and to make predictions about such populations based on estimates of its mean and standard deviation. It represents the way many real life populations are distributed.

The Normal Distribution is the most commonly used and abused distribution in statistics and serves as the foundation of many Six Sigma and Quality Control statistical tools. It is the most recognized distribution in statistics.The shape of the distribution is a function of two parameters, the mean and the standard deviation.   

Many things are normally distributed, or very close to it. The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work well even when the distribution is not normal.

For example, there's a very strong relationship between the size of a sample N and the extent to which a sampling distribution approaches the normal form.

Many sampling distributions based on large N can be approximated by the normal distribution even though the population distribution itself is definitely not normal. 

Characteristics of the Normal Distribution

  • The mean, median, and mode of a normal distribution are equal. 
  • The area under the normal curve is equal to 1.0. 
  • Normal distributions are denser in the center and less dense in the tails. 
  • Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ). 
  • Only random error is present
  • The process is free of assignable causes.
  • The process is free of Drift and Shift.


The Normal DistributionNormal Distribution


A. Population’s mean - Central tendency of the distribution. The area under the curve is split half on either side of this point.

B. Population’s standard deviation - Measure of dispersion of the distribution given by the horizontal distance between the mean and the point of inflection of the curve.

C. Normal curve - Bell shaped curve representing the distribution. This curve is asymptotic to the horizontal axis.

D. Probability - The area under the curve represents the probability of occurrence of an event. The total area from - to + is equal to one (1).

E. Horizontal axis - Axis of measure of a variable such as a CT characteristic.

F. The probability that the value of X lies between points X1and X2 is equal to the area under the curve from point X1 to point X2.

G. Probability Function of the normal distribution.

Major Considerations

A sample that is representative of the population shall be used to calculate an estimate of the population’s mean and standard deviation.

For a more common and easier way to calculate probabilities using the standard normal distribution, see the “Standard Normal Deviate (Z)”.

Application Cookbook

1. Calculate an estimate of the population’s mean and standard deviation.

2. Considering the specification limits of the particular case, determine the values of the independent variable for which you want to calculate the probability of occurrence of an event. Usually, this is the point beyond which a defect will occur.

3. Use Excel functions to calculate the normal cumulative distribution and the inverse of the normal cumulative distribution.

4. In Minitab use the following menu to generate the distribution. CALC>PROBABILITY DISTRIBUTIONS>NORMAL


Non-normal Distribution

When you have a Non-normal Distribution there is usually special or more obvious causes of variation that can be readily apparent upon process investigation.

Special Cause: Variation that is caused by known factors that result in a non-random distribution of output. Also referred to as “Assignable Cause” variation. This variation

Common Cause: Variation caused by unknown factors resulting in a steady but random distribution of output around the average of the data. It is the variation left over after Special Cause variation has been removed and typically (not always) follows a Normal Distribution.

If we know that the basic structure of the data should follow a Normal Distribution, but plots from our data shows otherwise; we know the data contain Special Causes.

While many processes in industry and nature behave according to the Normal Distribution, many processes in business, particularly in the areas of service and transactions, do not.

Normal Data are not common in the transactional world. There are lots of meaningful statistical tools you can use to analyze your data. It just means you may have to think about your data in a slightly different way.

Empirical Rule

The area under the curve between any two points represents the proportion of the distribution.  The concept of determining the proportion between 2 points under the standard Normal curve is a critical component to estimating Process Capability

Normal Distribution - Distance Between Two Points

Normal Distribution Distance Between Two PointsThe distance between any two points can be calculated.

The Empirical Rule allows you predict or more appropriately make estimates on how your process is performing. 

  • 68.27 % of the data will fall within +/- 1 Standard Deviation
  • 95.45 % of the data will fall within +/- 2 Standard Deviations
  • 99.73 % of the data will fall within +/- 3 Standard Deviations
  • 99.9937 % of the data will fall within +/- 4 Standard Deviations
  • 99.999943 % of the data will fall within +/- 5 Standard Deviations
  • 99.9999998 % of the data will fall within +/- 6 Standard Deviations

Normal Distribution Empirical Rule

Normal Distribution Empirical RuleEmpirical Rule

From Normal Distribution to Six Sigma Tools.

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