Lognormal Distribution
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Lognormal Distribution |
If a data set is known to follow a lognormal distribution, transforming the data by
taking a logarithm yields a data set that is normally distributed. This is shown in the
table below.
The most common transformation is made by taking the natural logarithm, but any base logarithm, such as base 10 or base 2, also yields a normal distribution. The remaining discussion will use the natural logarithm denoted as "ln". When random variables are summed, as the sample size increases, the distribution of the sum becomes a normal distribution regardless of the distribution of the individuals. Since lognormal random variables are transformed to normal random variables by taking the logarithm, when random variables are multiplied, as the sample size increases, the distribution of the product becomes a lognormal distribution regardless of the distribution of the individuals. This is because the logarithm of the product of several variables is equal to the sum of the logarithms of the individuals. This is shown below. 
The lognormal probability density function is
where μ is the location parameter or log mean, and σ is the scale parameter or log standard deviation. The location parameter is the mean of the data set after transformation by taking the logarithm, and the scale parameter is the standard de viation of the data set after transformation.The lognormal distribution takes on several shapes depending on the value of the shape parameter. The lognormal distribution is skewed right, and the skewness increases as the value of σ increases. This is sh own in the figure below.
The lognormal cumulative distribution and reliability functions are
where F(x) is the standard normal cumulative distribution function. Example: The data below is the time to fail for 4 light bulbs, and is known to have a lognormal distribution. What is the reliability at 100 hours? 115 hours 155 hours Solution: The table below shows this data and the transformation to normal.
The parameters of the lognormal distribution are found by computing the mean and standard deviation of the transformed data in the second column of the table above. The mean of the transformed data is
The sum of the second column of the table above is 20.3777. If each value in the second column of the table above is squared, the sum of the squared values is 104.0326. Using these values, the sample standard deviation of the values in the second column of the table above is
An easier method of computing the standard deviation is to use the Microsoft Excel function =stdev(). The reliability at 100 hours is
From a standard normal table, Microsoft Excel or Lotus 123, the standard normal cumulative distribution function at z = -1.807 is 0.0354, thus the reliability at 100 hours is R(100) = 1-0.0354 = 0.9646 The reliability at 100 hours can also be computed using the Microsoft Excel function =1-NORMDIST(LN(100),5.0944,0.2708,1) Click Here to download this solution in Microsoft Excel. The location parameter, or log mean, is often mistaken for the mean of the lognormal distribution. The mean of the lognormal distribution can be computed from its parameters
The variance of the lognormal distribution is
The lognormal hazard function has a unique behavior; it increases initially, then decreases and eventually approaches zero. This means that items with a lognormal distribution have a higher chance of failing as they age for some period of time, but after survival to a specific age, the probability of failure decreases as time increases. The lognormal hazard function is shown in the figure below.
In the figure above, when σ = 2, the hazard function increases so quickly that it cannot be seen on the graph.
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