Normal approximation to the Lognormal distribution


When the Lognormal distribution lognormal(m, s) has an arithmetic mean m that is much larger than its arithmetic standard deviation s, the distribution tends to look like a Normal(m, s), i.e.:

Lognormal(m, s) » Normal(m, s)

A general rule of thumb for this approximation is m > 6s. This approximation is not really useful from the point of view of simplifying the mathematics, but it is helpful in being able to quickly think of the range of the distribution and its peak in such circumstances. For example, we know that 99.7% of a Normal distribution is contained within a range +/- 3s  from the mean m. So, for a Lognormal(15, 2), we would estimate the distribution to be almost completely contained within a range [9,21] and that it peaks at a little below 15 (remember that the mode, median and mean appear in that order from left to right for a right skewed distribution).

 

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