scipy.stats.f = <scipy.stats._continuous_distns.f_gen object>[source]#

An F continuous random variable.

For the noncentral F distribution, see ncf.

As an instance of the rv_continuous class, f 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.

See also



The F distribution with \(df_1 > 0\) and \(df_2 > 0\) degrees of freedom is the distribution of the ratio of two independent chi-squared distributions with \(df_1\) and \(df_2\) degrees of freedom, after rescaling by \(df_2 / df_1\).

The probability density function for f is:

\[f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}} {(df_2+df_1 x)^{(df_1+df_2)/2} B(df_1/2, df_2/2)}\]

for \(x > 0\).

f accepts shape parameters dfn and dfd for \(df_1\), the degrees of freedom of the chi-squared distribution in the numerator, and \(df_2\), the degrees of freedom of the chi-squared distribution in the denominator, respectively.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, f.pdf(x, dfn, dfd, loc, scale) is identically equivalent to f.pdf(y, dfn, dfd) / 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.


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

Calculate the first four moments:

>>> dfn, dfd = 29, 18
>>> mean, var, skew, kurt = f.stats(dfn, dfd, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(f.ppf(0.01, dfn, dfd),
...                 f.ppf(0.99, dfn, dfd), 100)
>>> ax.plot(x, f.pdf(x, dfn, dfd),
...        'r-', lw=5, alpha=0.6, label='f 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 = f(dfn, dfd)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = f.ppf([0.001, 0.5, 0.999], dfn, dfd)
>>> np.allclose([0.001, 0.5, 0.999], f.cdf(vals, dfn, dfd))

Generate random numbers:

>>> r = f.rvs(dfn, dfd, 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)


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

Random variates.

pdf(x, dfn, dfd, loc=0, scale=1)

Probability density function.

logpdf(x, dfn, dfd, loc=0, scale=1)

Log of the probability density function.

cdf(x, dfn, dfd, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, dfn, dfd, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, dfn, dfd, loc=0, scale=1)

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

logsf(x, dfn, dfd, loc=0, scale=1)

Log of the survival function.

ppf(q, dfn, dfd, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, dfn, dfd, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, dfn, dfd, loc=0, scale=1)

Non-central moment of the specified order.

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

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

entropy(dfn, dfd, 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=(dfn, dfd), 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(dfn, dfd, loc=0, scale=1)

Median of the distribution.

mean(dfn, dfd, loc=0, scale=1)

Mean of the distribution.

var(dfn, dfd, loc=0, scale=1)

Variance of the distribution.

std(dfn, dfd, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, dfn, dfd, loc=0, scale=1)

Confidence interval with equal areas around the median.