Log-normal Distribution

Definition 1: A random variable x is log-normally distributed provided the natural log of x, ln x, is normally distributed. See Exponentials and Logs and Built-in Excel Functions for a description of the natural log.

Observation: As discussed in Transformations, sometimes it is useful to use a transformation of the population being studied. In particular, since the normal distribution has very desirable properties, transforming a random variable into a variable that is normally distributed by taking the natural log can be useful.

Note that the log-normal distribution is not symmetric, but is skewed to the right. If you have data that is skewed to the right that fits the log-normal distribution, you may be able to access various tests described elsewhere in this website that require data to be normally distributed.

Excel Functions: Excel provides the following two functions:

LOGNORMDIST(x, μ, σ) = NORMDIST(LN(x), μ, σ, TRUE)

LOGINV(p, μ, σ) is the inverse of LOGNORMDIST(x, μ, σ); i.e. LOGINV(p, μ, σ) = the value x such that LOGNORMDIST(x, μ, σ) = p

Excel 2010/2013 provide the following additional functions: LOGNORM.DIST(x, μ, σ, cum) where cum takes the values TRUE or FALSE and LOGNORM.DIST(x, μ, σ, cum) = NORM.DIST(LN(x), 0, 1, cum), and LOGNORM.INV which is equivalent to LOGINV.

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