# Log-normal Distribution

Definition 1: A random variable x is log-normally distributed provided the natural log of x, ln x, is normally distributed. The probability density function (pdf) of the log-normal distribution is

Observation: Some key statistical properties are:

Observation: 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.

Figure 1 shows a chart of the log-normal distribution with mean 0 and standard deviations 1, .5 and .25.

Figure 1 – Chart of Log-normal Distribution

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:

LOGNORM.DIST(xμσ, cum) = the log-normal cumulative distribution function with mean μ and standard deviation σ at x if cum = TRUE and the probability density function of the log-normal distribution if cum = FALSE.

LOGNORM.INV(p, μ, σ) = the inverse of LOGNORM.DIST(x, μσ, TRUE)

Note that:

LOGNORM.DIST(xμ, σ, TRUE) = NORM.DIST(LN(x), μ, σ, TRUE)

LOGNORM.DIST(xμ, σ, FALSE) = NORM.DIST(LN(x), μ, σ, FALSE)/x

LOGNORM.INV(pμ, σ) = EXP(NORM.INV(pμ, σ))

These functions are not available in versions of Excel prior to Excel 2010. Instead, these versions of Excel use LOGNORMDIST(xμ, σ), which is equivalent to LOGNORM.DIST(xμ, σ, TRUE), and LOGINV(p, μ, σ), which is equivalent to LOGNORM.INV(p, μ, σ).  For the pdf function, the formula equivalent to LOGNORM.DIST(xμ, σ, FALSE) is NORMDIST(LN(x), μ, σ, FALSE)/x.

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