scipy.stats.lognorm = <scipy.stats._continuous_distns.lognorm_gen object>[source]#

A lognormal continuous random variable.

As an instance of the rv_continuous class, lognorm object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The probability density function for lognorm is:

\[f(x, s) = \frac{1}{s x \sqrt{2\pi}} \exp\left(-\frac{\log^2(x)}{2s^2}\right)\]

for \(x > 0\), \(s > 0\).

lognorm takes s as a shape parameter for \(s\).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, lognorm.pdf(x, s, loc, scale) is identically equivalent to lognorm.pdf(y, s) / scale with y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

Suppose a normally distributed random variable X has mean mu and standard deviation sigma. Then Y = exp(X) is lognormally distributed with s = sigma and scale = exp(mu).


>>> import numpy as np
>>> from scipy.stats import lognorm
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> s = 0.954
>>> mean, var, skew, kurt = lognorm.stats(s, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(lognorm.ppf(0.01, s),
...                 lognorm.ppf(0.99, s), 100)
>>> ax.plot(x, lognorm.pdf(x, s),
...        'r-', lw=5, alpha=0.6, label='lognorm pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = lognorm(s)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = lognorm.ppf([0.001, 0.5, 0.999], s)
>>> np.allclose([0.001, 0.5, 0.999], lognorm.cdf(vals, s))

Generate random numbers:

>>> r = lognorm.rvs(s, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)

The logarithm of a log-normally distributed random variable is normally distributed:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> fig, ax = plt.subplots(1, 1)
>>> mu, sigma = 2, 0.5
>>> X = stats.norm(loc=mu, scale=sigma)
>>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
>>> x = np.linspace(*X.interval(0.999))
>>> y = Y.rvs(size=10000)
>>> ax.plot(x, X.pdf(x), label='X (pdf)')
>>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
>>> ax.legend()


rvs(s, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, s, loc=0, scale=1)

Probability density function.

logpdf(x, s, loc=0, scale=1)

Log of the probability density function.

cdf(x, s, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, s, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, s, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, s, loc=0, scale=1)

Log of the survival function.

ppf(q, s, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, s, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, s, loc=0, scale=1)

Non-central moment of the specified order.

stats(s, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(s, loc=0, scale=1)

(Differential) entropy of the RV.


Parameter estimates for generic data. See for detailed documentation of the keyword arguments.

expect(func, args=(s,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)

Expected value of a function (of one argument) with respect to the distribution.

median(s, loc=0, scale=1)

Median of the distribution.

mean(s, loc=0, scale=1)

Mean of the distribution.

var(s, loc=0, scale=1)

Variance of the distribution.

std(s, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, s, loc=0, scale=1)

Confidence interval with equal areas around the median.