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