The F Distribution

Statistics • Analysis of Variance

Written by STEM Calculators Team Published December 26, 2025 Updated February 24, 2026

Choose a task, enter the degrees of freedom, and provide either an F value or a right-tail area α. The F distribution is defined only for F > 0 and is typically used with right-tail areas.

Task

Tip: a left-tail critical value can also be found using the reciprocal relation.

df (numerator)

df (denominator)

F value (x)

Used for the probability tasks.

Right-tail area (α)

Used for the critical-value tasks.

Graph scale (suggested percentile)

Controls how far right the graph extends.

Batch mode (paste CSV or upload)

CSV columns: task, df1, df2, x, alpha.
Task values: rtail, cdf, crit, leftcrit.

Paste CSV

Upload CSV file

The uploaded file will be loaded into the paste box above.

Batch output: a table + downloadable CSV

Precision: probabilities shown to ~3 significant figures

Ready

Output: —

Enter values and click “Calculate”.

F distribution graph

A probability region will be shaded after calculation.

df = (—, —)

Labels are drawn in the current theme color; the shaded region matches the selected task (right tail or left/cumulative).

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The F Distribution

The F distribution is a continuous distribution defined on F > 0 and is typically skewed to the right. It has two degrees of freedom: df for the numerator and df for the denominator, written as (df₁, df₂).

Key formulas

CDF (left area). The cumulative probability can be computed using a regularized incomplete beta function.

\[ \begin{aligned} t &= \frac{df_1 \cdot x}{df_1 \cdot x + df_2} \\ P(F \le x) &= I_{t}\!\left(\frac{df_1}{2},\,\frac{df_2}{2}\right) \end{aligned} \]

Right tail. Many table lookups and hypothesis tests use a right-tail area.

\[ \begin{aligned} P(F \ge x) &= 1 - P(F \le x) \end{aligned} \]

PDF (shape). The probability density function is:

\[ \begin{aligned} f(x) &= \frac{\left(\frac{df_1}{df_2}\right)^{df_1/2}\,x^{df_1/2 - 1}} {B\!\left(\frac{df_1}{2},\,\frac{df_2}{2}\right)\left(1 + \frac{df_1}{df_2}\,x\right)^{(df_1+df_2)/2}}, \quad x>0 \end{aligned} \]

How to use the calculator

Right-tail area: enter df values and an x value to compute P(F ≥ x).
Cumulative area: enter df values and an x value to compute P(F ≤ x).
Critical value (right tail): enter df values and α to find x such that P(F ≥ x) = α.
Left critical value: enter df values and α to find x such that P(F ≤ x) = α.

Reciprocal relation (useful for left tails)

If F has degrees of freedom (df₁, df₂), then 1/F follows an F distribution with swapped degrees of freedom. This gives a convenient connection between left and right tails.

\[ \begin{aligned} F &\sim F(df_1, df_2) \\ \frac{1}{F} &\sim F(df_2, df_1) \end{aligned} \]

Frequently Asked Questions

What is the F distribution used for?

The F distribution is a continuous distribution defined for F > 0 with two degrees of freedom (df1, df2). It is commonly used in ANOVA and other variance ratio tests where test statistics follow an F model.

What is the difference between P(F >= x) and P(F <= x)?

P(F <= x) is the cumulative (left-tail) probability up to x, while P(F >= x) is the right-tail probability beyond x. For the same df1 and df2, the right-tail area equals 1 minus the cumulative area.

How do I find an F critical value for a right-tail test?

Choose the critical value task, enter df1 and df2, and enter alpha as the right-tail area. The calculator returns x such that P(F >= x) = alpha.

What do df1 and df2 mean in the F distribution?

df1 is the numerator degrees of freedom and df2 is the denominator degrees of freedom. Together they control the shape of the F curve and the location of critical values and tail areas.

How can I get a left-tail critical value for the F distribution?

Use the left critical value task and enter alpha as the left-tail area P(F <= x) = alpha. A useful identity is that if F ~ F(df1, df2), then 1/F ~ F(df2, df1), which connects left and right tails.

Calculation steps

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Frequently Asked Questions

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