Non-Normal Distributions - Lognormal Distribution
The log-normal distribution has the probability density function:
where
| μ | is the location parameter |
| σ | is the scale parameter |
| e | is the base of the natural logarithm, sometimes called Euler's e (2.71...) |
Threshold (location) parameter
The valid range for the log-normal distribution is from 0 to infinity. Instead of 0 (zero), Statistica allows you to enter a different value for the lowest threshold (location) parameter; that value will be subtracted from the data values before the log-normal distribution is fitted.
Thus, the threshold value must be less than the smallest observed value.
Applications
The log-normal distribution results if the logarithm of a random variable follows the normal distribution.
Since the normal distribution can be thought of as the result of adding a large number of independent errors, the log-normal distribution can be thought of as the result of multiplying a large number of independent errors (adding the log-errors). Thus, for example, if the measured quality characteristic of an item depends on a large number of random impulses that occur over time, and that have an impact that is proportional to the time at which they occur, then the resulting distribution of the quality characteristic will be log-normal. As an example (and metaphor) from biology, consider the size of an organism which is the result of a continuous growth process. If the growth rate at each developmental stage is affected by a large number of independent random factors, then the resultant distribution of the size of the organism will follow the log-normal distribution.
Estimation
Maximum likelihood parameter estimates for the log-normal distribution are computed by transformation of the normal distribution (see Evans, Hastings, and Peacock, 1993).