Lognormal2(m,s) - no Crystal Ball distribution = EXP(Normal(m,s))

Lognormal equations (format 2)

One usually defines the Lognormal distribution in terms of its mean and standard deviation, as in the Lognorm distribution.

The definition of a Lognormally distributed random variable is that the log of that variable is Normally distributed. Thus, an alternative way of defining a Lognormally distributed variable is to specify the mean and standard deviation of the corresponding Normal distribution, which is the Lognorm2 distribution. Therefore:

EXP(Normal(m,s)) = Lognorm2(m,s)

Note that we normally used the natural log, rather than base 10, i.e.:

EXP(Normal(m,s)) = Lognorm2(m,s)

10^(Normal(m,s)) ≠ Lognorm2(m,s)

#### Uses

Taking logs might seem like an unnecessary complication, but many people whose work involves statistics and who consider Lognormal random variables, habitually record and discuss the mean and the standard deviation of the corresponding Normal distribution, for example in experiments involving the growth of bacteria. They do this because variables, like the number of bacteria, can have wildly varying means and standard deviations between experiments, and taking logs can make comparisons much easier. Most people would usually use 10 as the log base rather than the natural exponent e but an easy conversion can be made between logs with the following identity:

Log_e(x)=Log_{10}(x)^*Log_e(10) |

for any positive value x. So if you have Log10 data you can take the mean (x10) and standard deviation (s10) of that log data and use it in Lognorm2 distribution above as follows:

= Lognorm2(m,s)~where~m=LN(10)^*\overline{x}_{10}~and~s=LN(10)*s_{10} |

= Lognorm2(LN(10)^*\overline{x}_{10},LN(10)*s_{10}) |

Note that this equation will return un-logged values of the variable.

For specific uses of the Lognormal distribution, see here.

#### Comments

The Excel function LOGNORMDIST(x,m,s) returns the cumulative distribution function for a Lognorm2(m,s).