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The Snedecor's $F$-distribution or the Fisher-Snedecor distribution (after Sir Ronald A. Fisher and George W. Snedecor) or short the $F$-distribution is a continuous probability distribution with range $[0, +\infty)$, depending on two parameters denoted $v_1, v_2$ (Lovric 2011). In statistical applications, $v_1, v_2$ are positive integers.

Let $Y_1$ and $Y_2$ be two independent random variables distributed as chi-square, with $v_1$ and $v_2$ degrees of freedom, respectively. Then the distribution of the ratio ($Z$)

$$Z = \frac{Y_1/v_1}{Y_2/v_2}$$

is called the $F$-distribution with $v_1$ and $v_2$ degrees of freedom. The $F$-distribution is often referred to as the distribution of the variance ratio (Lovric 2011).

A $F$-distribution has two numbers of degrees of freedom, $v_1$ and $v_2$, determining its shape. The first number of degrees of freedom, $v_1$, is called the degrees of freedom of the numerator and the second, $v_2$, the degrees of freedom of the denominator.

start = 0
end = 3

v1 = (1, 5, 10, 20, 50, 100)
v2 = (100, 50, 20, 10, 5, 1)

col_map = plt.get_cmap("Spectral", len(v1))

handle = ["$v_1=1$", "$v_1=5$", "$v_1=10$", "$v_1=20$", "$v_1=50$", "$v_1=100$"]


plt.figure(figsize=(10, 5))
plt.ylim([-0.1, 1.5])

x = np.arange(0, 3, 0.001)

plt.title(
    "Densities of F-distribution with $v_1 = 1, 5, 10, 20, 50, 100$ and $v_2 = 20$",
    fontsize=16,
)

for d, c, h in zip(v1, col_map(range(len(v1))), handle):
    y = stats.f.pdf(x, dfn=d, dfd=20)
    plt.plot(x, y, color=c, label=h)

plt.axvline(x=0, color="grey", linewidth=0.9)
plt.axhline(y=0, color="grey", linewidth=0.9)
plt.legend(fontsize=14)
plt.show()

start = 0
end = 3

v1 = (1, 5, 10, 20, 50, 100)
v2 = (100, 50, 20, 10, 5, 1)

col_map = plt.get_cmap("Spectral", len(v1))

handle = ["$v_2=1$", "$v_2=5$", "$v_2=10$", "$v_2=20$", "$v_2=50$", "$v_2=100$"]


plt.figure(figsize=(10, 5))
plt.ylim([-0.1, 1.5])

x = np.arange(0, 3, 0.001)

plt.title(
    "Densities of F-distribution with $v_1 = 20$ and $v_2 = 1, 5, 10, 20, 50, 100$",
    fontsize=16,
)

for d, c, h in zip(v1, col_map(range(len(v1))), handle):
    y = stats.f.pdf(x, dfn=20, dfd=d)
    plt.plot(x, y, color=c, label=h)

plt.axvline(x=0, color="grey", linewidth=0.9)
plt.axhline(y=0, color="grey", linewidth=0.9)
plt.legend(fontsize=14)
plt.show()

Basic Properties of $F$-Curves (Weiss, 2010)

• The total area under an $F$-curve is equal to 1.
• An $F$-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so.
• An $F$-curve is right skewed.

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Citation

The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.