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Assessing the Lognormal Distribution

The Lognormal distribution is widely used in reliability studies. Consequently, there is a strong interest in assessing whether a data set comes from such distribution. The results in the previous sections show how this is now easy to do. When a random variable (e.g., device life) is distributed Lognormal, the Logarithm (base e) of the random variable (e.g., Log life) is distributed Normal. This property carries on to data sets. When a data set comes from a Lognormal population, then the Logarithm of these data are distributed as a Normal.

In practice, to assess the Lognormality of a data set, we take the Logarithms of the original data and assess the Normality of the transformed data, as done in the sections above. For example, the data set in Table 6 comes from a Lognormal distribution with Location parameter 4 and Scale parameter 0.4.

Table 6. Original Data (Lognormal)
 67.842 91.03 42.974 42.849 46.459 64.746 55.031 38.326 119.612 62.903 31.778 87.068 58.854 44.79 69.054 69.222 39.334 121.592 90.537 99.651 93.44 31.021 47.152 63.716 92.824 36.03 104.526 62.006 35.605 35.019 32.102 24.288 80.42 132.861 48.886 57.911 79.527 37.659 63.223 110.359 77.153 84.713 52.391 42.475 65.333

We then transform these data by taking Logarithms of each element. For the first data point: Log(67.842) = 4.21718. The transformed data are shown in Table 7.

Table 7. Transformed Data (LogELN)
 4.21718 4.51119 3.7606 3.75769 3.83858 4.17048 4.0079 3.64612 4.78425 4.14159 3.45877 4.46668 4.07506 3.80198 4.23489 4.23732 3.67210 4.80067 4.50576 4.60168 4.53732 3.43466 3.85338 4.15443 4.53071 3.58434 4.64944 4.12723 3.57250 3.55589 3.46892 3.18999 4.38726 4.88930 3.88949 4.05891 4.3761 3.62857 4.14667 4.70374 4.34579 4.43927 3.95874 3.74891 4.17949

Descriptive statistics for both the Lognormal data set and its Logarithmic transformation are given in Table 8. For example, results for the sample Mean: Ln(65.21) = 4.177 ≈ 4.091. We now assess the Normality of the transformed data set (LogELN) by repeating the work discussed in the previous sections. If these transformed data fulfill the Properties given in Table 5, then the original data (Table 6) are distributed Lognormal.

Table 8. Descriptive Statistics for the Data and their Normal Transformation
Statistics Lognormal LogELN
N 45 45
Mean 65.21 4.0911
Median 62.90 4.1416
StDev 27.36 0.4244
Min 24.29 3.190
Max 132.86 4.889
Q1 42.66 3.753
Q3 85.89 4.453

These empirical results help assess the plausibility of the Normality or the Lognormality assumptions of a given life data set. If, at such point, a stronger case for the validity of these distributions is required, then a number of theoretical GoF tests can be carried out.