Lognormal Distribution Percent Point (Quantile) Calculator

Find the percent point x=exp(mu+sigma*z) of a lognormal distribution from probability p, a tail method, mu and sigma. Shows the matching z, cumulative probability, mean, median and variance, plus PDF and CDF graphs.

Input

Enter probability p, a method, and the log scale mean mu and standard deviation sigma to compute the lognormal percent point x=exp(mu+sigma*z).

Probability p

A value greater than 0 and less than 1 (e.g. 0.95)

Method

Finds x such that P(X at most x)=p.

Log mean mu

Mean of log X (any real number)

Log standard deviation sigma

Standard deviation of log X (positive)

Result

Percent point x for probability p=0.95

5.1802516

Standard normal quantile z

1.64485363

Cumulative probability P(X at most x)

0.95

Median

Mean

1.64872127

Variance

4.67077427

Probability density (PDF) and cumulative distribution (CDF)

Density PDF

Cumulative CDF

Percent point x

How it works

In a lognormal distribution, log X follows a normal distribution N(mu, sigma^2). The percent point is the value x matching a probability p, given by x=exp(mu+sigma*z), where z is the standard normal quantile.
Three methods are available. Lower means P(X at most x)=p, upper means P(X at least x)=p, and two sided corresponds to the upper bound of the central interval that holds probability p.
The standard normal quantile z is computed from the inverse cumulative distribution function (probit) using a rational polynomial approximation refined by Newton iteration for high accuracy.
The median is exp(mu), the mean is exp(mu+sigma^2/2), and the variance is (exp(sigma^2)-1)*exp(2*mu+sigma^2). Note that mu and sigma are the mean and standard deviation of the underlying normal distribution after taking logs.
Enter sigma as a positive value and probability p strictly between 0 and 1.

Reviews

Tell us what you think of this calculator.