You can use our calculators to calculate the probability density function (PDF) or cumulative distribution function (CDF) of the F-distribution

F Distribution Calculator

F Distribution Calculator

For degrees of freedom \(d_1>0\) and \(d_2>0\), the PDF is:

$$ f(x;d_1,d_2)=\frac{\Gamma\Bigl(\frac{d_1+d_2}{2}\Bigr)}{\Gamma\Bigl(\frac{d_1}{2}\Bigr)\Gamma\Bigl(\frac{d_2}{2}\Bigr)}\Bigl(\frac{d_1}{d_2}\Bigr)^{\frac{d_1}{2}}x^{\frac{d_1}{2}-1}\Bigl(1+\frac{d_1}{d_2}x\Bigr)^{-\frac{d_1+d_2}{2}}, \quad x>0. $$

and the CDF is:

$$ F(x;d_1,d_2)=I_{\frac{d_1x}{d_1x+d_2}}\Bigl(\frac{d_1}{2},\frac{d_2}{2}\Bigr), \quad x>0. $$

Mean exists if \(d_2>2\) and variance if \(d_2>4\).

Step 1: Enter Parameters

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Enter a number greater than 0 (e.g., 1)

F Distribution: $$ f(x;d_1,d_2)=\frac{\Gamma\Bigl(\frac{d_1+d_2}{2}\Bigr)}{\Gamma\Bigl(\frac{d_1}{2}\Bigr)\Gamma\Bigl(\frac{d_2}{2}\Bigr)}\Bigl(\frac{d_1}{d_2}\Bigr)^{\frac{d_1}{2}}x^{\frac{d_1}{2}-1}\Bigl(1+\frac{d_1}{d_2}x\Bigr)^{-\frac{d_1+d_2}{2}},\quad x>0. $$

The F distribution is a probability distribution used to compare variances and means. It's used in hypothesis testing, analysis of variance (ANOVA), and regression analysis. 

What it's used for

  • Hypothesis testing: Used to determine if two population variances are equal 
  • ANOVA: Used to determine if the differences between sample means are statistically significant 
  • Regression analysis: Used to test the overall significance of a regression model 

How it's used

  • The F-statistic is calculated as the ratio of two variances 
  • The F-statistic is compared to the critical value from the F-distribution to determine if the null hypothesis should be rejected 

Characteristics

  • The F-distribution is a continuous probability distribution that's defined for an infinite number of values 
  • It's a right-skewed distribution with a minimum value of 0 and no maximum value 
  • The F-distribution is also known as the Fisher-Snedecor distribution 

Other uses

The F-distribution is used in situations where the χ2 and the student's t distributions are not appropriate

When to use F Distribution

You use an F distribution when you want to compare two variances or more than two means. It's also used to determine confidence intervals and critical regions for hypothesis tests. 

When to use 

  • To compare two variances
  • To compare more than two means
  • To determine if two populations have equal variances
  • To test the validity of a multiple regression equation
  • To analyze the results of an ANOVA

How it works

  • The F distribution is a probability distribution that's positively skewed 
  • The shape of the F distribution depends on the degrees of freedom for the numerator and denominator 
  • The F distribution is the ratio of two independent chi-squared distributions 
  • The F distribution is used in hypothesis testing to determine whether two population variances are equal 

Example 

  • A botanical research team might use an F distribution to determine if there is any improvement in plant growth after six months

F distribution example

The F distribution is a statistical distribution used to compare variances and test hypotheses. It's often used in analysis of variance (ANOVA). 

Examples

  • Comparing gas mileage: A consumer might compare the average gas mileage of several car models. 
  • Comparing weight loss: A researcher might test three different diet plans for mean weight loss. 
  • Comparing pollution levels: An environmentalist might compare the average amount of pollution in several bodies of water. 
  • Comparing income: A sociologist might compare the amount of income a person earns based on their upbringing. 

Key characteristics

  • The F distribution is skewed to the right. 
  • There is a different curve for each set of degrees of freedom in the numerator and denominator. 
  • The F statistic is a fraction, and its value is greater than or equal to zero. 
  • Larger F ratio values signal that the effect being studied may have statistical significance. 

Related terms F-test, F-ratio, Degrees of freedom, and Hypothesis test. 

You can use an F-distribution table to find F-values. 

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