This is just an Excerpt from a larger document, click here to view the entire document.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
91.030
42.974
42.849
46.459
64.746
55.031
38.326
119.612
62.903
31.778
87.068
58.854
44.790
69.054
69.222
39.334
121.592
90.537
99.651
93.440
31.021
47.152
63.716
92.824
36.030
104.526
62.006
35.605
35.019
32.102
24.288
80.420
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
3.76060
3.75769
3.83858
4.17048
4.00790
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
4.37610
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.
