Lognormal Distribution

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Lognormal Distribution


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

Purpose

The lognormal distribution is used to describe distributions associated to life spans, reaction times, income distributions, economic data, etc. 

It differs from the normal distribution in several ways. A major difference is in its shape: where the normal distribution is symmetrical, a lognormal one is not. Because the values in a lognormal distribution are positive, they create a right skewed curve.

Since many kinds of data in real life have a positively-skewed distribution, the lognormal distribution has been widely applied in many areas. For example, it's commonly used to model the lifetime of products whose failure modes are fatigue or stress in nature. This includes almost all mechanical systems.

Normal vs. Lognormal


The lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed.

In statistical analysis, a logarithmic transformation is often applied to a set of positively-skewed distributed data before proceeding with the analysis. It forces the skewed distribution to approximate the normal distribution. This approach works well for statistical analysis used is Six Sigma and Quality Engineering.

Both normal and lognormal distributions are used in statistical analysis to describe the probability of an event occurring.

Both normal and lognormal distributions are used in statistical analysis to describe the probability of an event occurring.

Flipping a coin is an easily understood example of probability. If you flip a coin 1000 times, what is the distribution of results? That is, how many times will it land on heads or tails? (Answer: half the time heads, the other half tails.)

This is a very simplified example to describe probability and the distribution of results. There are many types of distributions, one of which is the normal or bell curve distribution.

In a normal distribution 68% (34%+34%) of the results fall within one standard deviation and 95% (68%+13.5%+13.5%) fall within 2 standard deviations. At the center (the 0 point in the image above), the median, or the middle value in the set, the mode, the value that occurs most often, and the mean, the arithmetic average, are all the same.


Anatomy

Lofnormal Distribution

Terminology

A. Vertical axis - Scale of measure of the lognormal distribution.

B. Horizontal axis - Scale of measure of the independent variable.

C. Curve representing a lognormal distribution. A variable x has a lognormal distribution if logA(x) is normally distributed. Conversely, we can say that if y = Ax, where x has a normal distribution, then Y is said to have a lognormal distribution.


Major Considerations

Caution shall be exercised in calculating probabilities and making predictions. For example, if Y represents the life of a component and Y = Ax where X has a normal distribution, then one would want to make predictions on the average life of the system, not on the mean of the logarithm of Y. To solve this problem, consult the Data Transformations tool.

Application Cookbook

1. Use Excel functions to calculate the cumulative probability and the inverse of the lognormal cumulative distribution of x where ln(x) is normally distributed.

2. In Minitab, the following menu to generate a lognormal distribution. CALC>PROBABILITY DISTRIBUTIONS>LOGNORMAL

From Lognormal Distribution to Six Sigma Tools.

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