Fact-checked by Grok 2 weeks ago

F -distribution

The F-distribution, also known as the Fisher–Snedecor distribution, is a continuous probability distribution that arises as the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.^[1]^[2] It is defined such that if U follows a chi-squared distribution with r degrees of freedom and V follows a chi-squared distribution with s degrees of freedom, then F = \frac{U/r}{V/s} follows an F-distribution with parameters r (numerator degrees of freedom) and s (denominator degrees of freedom), denoted F(r, s).^[2]^[3] This distribution is right-skewed, with its shape determined by the values of r and s; it becomes more symmetric and approaches a normal distribution as s increases.^[1]^[2] Named after the British statistician and geneticist Sir Ronald A. Fisher (1890–1962) and the American statistician George Snedecor, the F-distribution was introduced in the early 1920s as part of Fisher's work on variance analysis in agricultural experiments at the Rothamsted Experimental Station.^[4]^[5] Fisher developed it to test the significance of differences in means across groups by comparing variances, building on his broader contributions to experimental design and statistical inference. Snedecor tabulated the distribution and named it "F" in Fisher's honor in the 1930s.^[4]^[6] A key property is its reciprocity: if W \sim F(r, s), then $1/W \sim F(s, r).^[2] The distribution has support on positive real numbers, but its mean is s/(s-2) for s > 2 and variance is \frac{2s^2(r+s-2)}{r(s-2)^2(s-4)} for s > 4.^[1] In statistical practice, the F-distribution underpins the F-test, which assesses the equality of variances from two normal populations by computing the ratio of sample variances.^[1]^[3] It is central to analysis of variance (ANOVA), where the F-statistic compares between-group variance to within-group variance to determine if observed differences in means are statistically significant.^[1]^[7] Applications extend to regression analysis for testing overall model significance and to confidence intervals for variance ratios.^[1]^[4] Tables and software compute critical values and p-values based on r and s, facilitating hypothesis testing in fields like biology, engineering, and social sciences.^[8]

Definition and Parameters

Probability Density Function

The probability density function (PDF) of the F-distribution, denoted F(d_1, d_2) where d_1 > 0 and d_2 > 0 are the numerator and denominator degrees of freedom, respectively, is given by

f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}} \cdot \frac{\Gamma\left(\frac{d_1 + d_2}{2}\right)}{\Gamma\left(\frac{d_1}{2}\right) \Gamma\left(\frac{d_2}{2}\right)}}{x}, \quad x > 0.

^[9]^[10] This formula can also be expressed in an equivalent form using the beta function B(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a + b)}, as

f(x; d_1, d_2) = \frac{\left( \frac{d_1}{d_2} \right)^{d_1/2} x^{d_1/2 - 1}}{B\left( \frac{d_1}{2}, \frac{d_2}{2} \right) \left( 1 + \frac{d_1}{d_2} x \right)^{(d_1 + d_2)/2}}, \quad x > 0,

which highlights the normalizing role of the beta function derived from the gamma functions.^[2] The gamma functions in the PDF serve as normalizing constants, originating from the PDFs of the underlying chi-squared distributions and ensuring the total probability integrates to 1 over (0, \infty).^[1] The F-distribution arises as the distribution of the random variable F = \frac{U / d_1}{V / d_2}, where U and V are independent chi-squared random variables with d_1 and d_2 degrees of freedom, respectively.^[10] To derive the PDF, start with the joint PDF of U and V, which is the product of their individual chi-squared PDFs: f_{U,V}(u,v) = f_U(u) f_V(v) = \frac{1}{2^{d_1/2} \Gamma(d_1/2)} u^{d_1/2 - 1} e^{-u/2} \cdot \frac{1}{2^{d_2/2} \Gamma(d_2/2)} v^{d_2/2 - 1} e^{-v/2} for u > 0, v > 0.^[11] Perform a change of variables to F = \frac{U / d_1}{V / d_2} and, say, W = V, yielding the Jacobian determinant |J| = \frac{d_1}{d_2} w. The joint PDF of F and W is then f_{F,W}(f, w) = f_{U,V}(u(f,w), w) |J|, where u = \frac{d_1}{d_2} f w. Integrating over w > 0 gives the marginal PDF of F, resulting in the beta integral form that simplifies to the expression involving gamma functions.^[11]^[10]

Support and Parameter Constraints

The support of the F-distribution consists of all positive real numbers, with the random variable F taking values in the interval (0, \infty). This reflects the fact that the distribution arises from the ratio of two positive quantities, ensuring no negative or zero values are possible under the standard parameterization. The probability density function is defined exclusively for x > 0, aligning with this support.^[1]^[12]^[10] The F-distribution is parameterized by two degrees of freedom, denoted d_1 and d_2, both of which must be strictly positive real numbers (d_1 > 0 and d_2 > 0) to ensure the distribution is well-defined and integrates to unity. While integer values are conventional in applications—stemming from their connection to sample-based variance estimates—the mathematical formulation permits non-integer parameters without loss of validity. For the mean to exist, d_2 > 2 is additionally required, and higher moments impose stricter conditions such as d_2 > 4 for the variance. In statistical testing scenarios, d_1 corresponds to the degrees of freedom associated with the numerator variance (often from the explained variation), while d_2 pertains to the denominator variance (typically from residual or unexplained variation).^[1]^[10]^[12] Limiting behaviors of the distribution occur as the parameters approach boundary values. As d_1 \to 0^+, the distribution concentrates near 0, with the probability density approaching 0 for all x > 0 but degenerating to a point mass at the origin in the limit. Conversely, as d_2 \to 0^+, the mass shifts toward infinity, rendering the density negligible across the support. When both d_1 and d_2 \to \infty, the F-distribution converges to a standard normal distribution, losing its characteristic right-skewness. These limits highlight the sensitivity of the distribution to parameter values near the boundaries of their allowable range.^[1]^[10]^[13]

Statistical Properties

Moments and Central Tendency

The expected value (mean) of a random variable F following the F-distribution with numerator degrees of freedom d_1 > 0 and denominator degrees of freedom d_2 > 0 is given by

\mathbb{E}[F] = \frac{d_2}{d_2 - 2}

provided that d_2 > 2; the mean is undefined otherwise, as the integral diverges.^[10] This expression arises from evaluating the first raw moment via the probability density function, which involves the beta function expressible in terms of gamma functions.^[14] The variance of F is

\mathrm{Var}(F) = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2 - 2)^2 (d_2 - 4)}

for d_2 > 4; it is undefined for smaller d_2, reflecting the heavy tails of the distribution.^[10] This formula is derived as the second central moment, using the mean and the second raw moment computed similarly through the density.^[14] Higher-order raw moments \mathbb{E}[F^k] for positive integer k exist only when d_2 > 2k, and are expressed using the gamma function as

\mathbb{E}[F^k] = \frac{\Gamma\left( \frac{d_1 + d_2}{2} - k \right) \Gamma\left( \frac{d_1}{2} + k \right)}{\Gamma\left( \frac{d_1 + d_2}{2} \right) \Gamma\left( \frac{d_1}{2} \right)} \cdot \left( \frac{d_2}{d_1} \right)^k.

This general formula, first derived systematically in the mid-20th century, allows computation of skewness, kurtosis, and other measures for specific d_1 and d_2.^[15] As d_1 and d_2 both approach infinity, the mean \mathbb{E}[F] converges to 1, indicating that the distribution concentrates around unity under large-sample conditions.^[10]

Shape Characteristics and Quantiles

The F-distribution exhibits a right-skewed shape, particularly when the denominator degrees of freedom d_2 are small, with the tail extending to the right and the bulk of the probability mass concentrated near zero. As both d_1 and d_2 increase, the distribution becomes more symmetric and approaches a normal distribution, reflecting the central limit theorem's influence on ratios of large-sample variances.^[9] The mode of the F-distribution, which is the value at which the probability density function achieves its maximum, is located at 0 when d_2 \leq 2. For d_2 > 2, the mode is given by

\frac{d_1 (d_2 - 2)}{d_2 (d_1 + 2)}.

This formula highlights how the modal value shifts rightward with increasing d_1 relative to d_2, influencing the distribution's peak position.^[9] Higher-order moments provide further insight into the shape. The skewness, measuring asymmetry, is positive and given by

\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2 - 4)}}{(d_2 - 6) \sqrt{d_1 (d_1 + d_2 - 2)}}

for d_2 > 6, indicating right-skewness that diminishes as d_2 grows.^[12] The excess kurtosis, quantifying tail heaviness relative to a normal distribution, is

$12 \frac{d_1 (5 d_2 - 22)(d_1 + d_2 - 2) + (d_2 - 4)(d_2 - 2)^2}{d_1 (d_2 - 6)(d_2 - 8)(d_1 + d_2 - 2)} - 3

for d_2 > 8, showing leptokurtosis (heavier tails) that decreases toward zero with larger degrees of freedom.^[12] These measures confirm the distribution's departure from normality for small parameters, with skewness and excess kurtosis both approaching 0 as d_1, d_2 \to \infty.^[9] Quantiles of the F-distribution are essential for determining critical values in hypothesis testing, where the upper-tail critical value F_{\alpha}(d_1, d_2) satisfies P(X > F_{\alpha}(d_1, d_2)) = \alpha for a random variable X \sim F(d_1, d_2). These quantiles lack closed-form expressions and are typically obtained from statistical tables or computational software. For large degrees of freedom, approximations facilitate quantile estimation; the Wilson-Hilferty transformation, originally developed for chi-squared distributions but applicable to F via its relation to chi-squared ratios, approximates the cube root of the variable as normally distributed, enabling efficient computation of tail probabilities and quantiles.^[9]^[16]

Derivations and Relationships

From Chi-Squared Distributions

The F-distribution with parameters d_1 and d_2 (degrees of freedom) is defined as the distribution of the ratio of two independent scaled chi-squared random variables. Specifically, if U \sim \chi^2_{d_1} and V \sim \chi^2_{d_2} are independent, then the random variable F = \frac{U / d_1}{V / d_2} follows an F-distribution, denoted F \sim F(d_1, d_2).^[2] This construction captures the distribution of variance ratios from samples drawn from normal populations, central to many statistical tests.^[1] To derive the probability density function of F from the chi-squared densities, apply the transformation of variables to the joint distribution of U and V. The probability density function of U is

f_U(u) = \frac{1}{2^{d_1/2} \Gamma(d_1/2)} u^{d_1/2 - 1} e^{-u/2}, \quad u > 0,

and similarly for V:

f_V(v) = \frac{1}{2^{d_2/2} \Gamma(d_2/2)} v^{d_2/2 - 1} e^{-v/2}, \quad v > 0.

Since U and V are independent, their joint density is f_{U,V}(u,v) = f_U(u) f_V(v) for u > 0, v > 0.^[17] Introduce the transformation F = \frac{U / d_1}{V / d_2} and S = V, which implies U = F \cdot (d_1 / d_2) \cdot S and S = V. The Jacobian matrix of this transformation from (U, V) to (F, S) has determinant with absolute value |J| = (d_1 / d_2) s. Thus, the joint density of F and S is

f_{F,S}(f,s) = f_U\left( f \cdot \frac{d_1}{d_2} \cdot s \right) f_V(s) \cdot \frac{d_1}{d_2} \cdot s, \quad f > 0, \, s > 0.

The marginal density of F is obtained by integrating out S:

f_F(f) = \int_0^\infty f_U\left( f \cdot \frac{d_1}{d_2} \cdot s \right) f_V(s) \cdot \frac{d_1}{d_2} \cdot s \, ds, \quad f > 0.

Substituting the chi-squared densities into this integral and performing the algebraic simplification—factoring constants, powers of f and s, and the exponential terms—yields an expression recognizable as a scaled beta density (or directly the F-density form after normalization). The integral evaluates to a gamma function ratio due to its resemblance to the gamma density kernel.^[11] This derivation hinges on the independence of U and V, which ensures the joint density factors and allows the transformation to proceed without correlation terms. For large d_1 and d_2, the central limit theorem implies that U / d_1 \approx N(1, 2/d_1) and V / d_2 \approx N(1, 2/d_2) approximately, so F behaves like the ratio of two independent normals centered at 1; this approximates the squared ratio of standard normals in the limit after suitable scaling, concentrating the distribution around 1.^[2] Ronald Fisher originally motivated the F-distribution through ratios of variances in experimental designs, such as comparing group variances under normality assumptions to test for equality in agricultural trials.^[1]

Links to t and Beta Distributions

The F-distribution with one degree of freedom in the numerator is directly related to the Student's t-distribution through a simple squaring transformation. Specifically, if T \sim t_{d_2}, then T^2 \sim F(1, d_2).^[18] This relationship arises because the t-distribution can be derived as the ratio of a standard normal variate to the square root of a chi-squared variate divided by its degrees of freedom, and squaring it yields the form of an F random variable with numerator degrees of freedom equal to 1. This link is particularly useful in statistical testing, where t-tests for means can be reframed in terms of F-tests for the special case of a single numerator degree of freedom. More generally, the F-distribution maintains a close algebraic connection to the beta distribution via a monotonic transformation that preserves moments and facilitates computational evaluation. If F \sim F(d_1, d_2), then the transformed variable B = \frac{d_1 F}{d_1 F + d_2} follows a beta distribution, B \sim \mathrm{Beta}\left( \frac{d_1}{2}, \frac{d_2}{2} \right)./05%3A_Special_Distributions/5.11%3A_The_F_Distribution) Conversely, starting from B \sim \mathrm{Beta}(\alpha, \beta), the variable F = \frac{\beta}{\alpha} \cdot \frac{B}{1 - B} follows an F-distribution with parameters F(2\alpha, 2\beta). This transformation is moment-preserving, as the first and higher moments of the F can be derived from those of the beta through the scaling and inversion. In the special case linking back to the t-distribution, when d_1 = 1, the transformation simplifies to B = \frac{F}{F + d_2} = \frac{T^2}{T^2 + d_2} \sim \mathrm{Beta}\left( \frac{1}{2}, \frac{d_2}{2} \right) for T \sim t_{d_2}, providing a direct bridge between all three distributions. These relationships are instrumental for numerical computation and quantile estimation, particularly since the cumulative distribution function (CDF) of the F-distribution can be expressed in terms of the regularized incomplete beta function: P(F \leq f) = I_{\frac{d_1 f}{d_1 f + d_2}} \left( \frac{d_1}{2}, \frac{d_2}{2} \right), where I_x(a, b) is the regularized incomplete beta function.^[14] Similarly, the CDF of the t-distribution involves the incomplete beta function, allowing efficient algorithms for tail probabilities and critical values to be implemented using beta routines, which are often more stable for integration in statistical software. This interconnectedness reduces computational complexity in deriving percentiles and supports approximations in large-sample scenarios. Asymptotically, when the denominator degrees of freedom d_2 \to \infty, the F-distribution F(d_1, d_2) converges in distribution to \chi^2_{d_1} / d_1, where \chi^2_{d_1} is a chi-squared random variable with d_1 degrees of freedom; this limit reflects the normalization of the denominator variance estimate approaching 1./05%3A_Special_Distributions/5.11%3A_The_F_Distribution) The ties to the beta and t-distributions further illuminate this behavior, as the beta transformation highlights the bounded nature of the scaled F variate, which approaches a degenerate form under the limit.

Applications in Inferential Statistics

F-Tests for Variance Equality

The F-test for equality of variances is a statistical procedure used to determine whether the variances of two independent populations are equal, based on samples drawn from each. The null hypothesis is typically stated as H_0: \sigma_1^2 = \sigma_2^2, where \sigma_1^2 and \sigma_2^2 are the population variances, while the alternative hypothesis for a two-sided test is H_a: \sigma_1^2 \neq \sigma_2^2. The test statistic is the ratio of the sample variances, F = \frac{s_1^2}{s_2^2}, where s_1^2 and s_2^2 are the sample variances from the first and second groups, respectively, and s_1^2 \geq s_2^2 by convention to ensure F \geq 1. Under the null hypothesis, assuming the populations are normally distributed, this statistic follows an F-distribution with degrees of freedom d_1 = n_1 - 1 and d_2 = n_2 - 1, where n_1 and n_2 are the sample sizes.^[19]^[20] For a two-tailed test at significance level \alpha, the rejection region consists of the upper and lower tails of the F-distribution: reject H_0 if F > F_{\alpha/2}(d_1, d_2) or if F < F_{1 - \alpha/2}(d_1, d_2), where F_{1 - \alpha/2}(d_1, d_2) = 1 / F_{\alpha/2}(d_2, d_1). The p-value is computed as twice the minimum of the cumulative distribution function (CDF) value at the observed F and 1 minus the CDF value, i.e., p = 2 \min \left( F_{\text{CDF}}(F; d_1, d_2), 1 - F_{\text{CDF}}(F; d_1, d_2) \right), with rejection if p < \alpha. Critical values can be obtained from F-distribution tables or software, referencing quantiles as defined in the shape characteristics of the F-distribution.^[19]^[20] The test assumes that the samples are independent and randomly drawn from normally distributed populations, with the normality condition being particularly critical for the validity of the F-distribution approximation. Violations of normality can lead to inflated Type I error rates, making the test sensitive to departures from the assumed distribution, especially in small samples. To address robustness issues, alternatives such as Levene's test, which transforms the data to absolute deviations from the group mean (or median for greater robustness) and applies an ANOVA-like F-test, are recommended when normality is questionable; this method, proposed by Levene in 1960, performs better under non-normal conditions.^[20]^[21]^[22] As a numerical illustration, consider testing the equality of variances in ceramic strength measurements from two batches of material, each with 240 observations. Batch 1 has a sample standard deviation of 65.55 and variance s_1^2 \approx 4297, while Batch 2 has a sample standard deviation of 61.85 and variance s_2^2 \approx 3826. The test statistic is F = \frac{4297}{3826} \approx 1.123, with degrees of freedom d_1 = d_2 = 239. For \alpha = 0.05, the upper critical value is F_{0.025}(239, 239) \approx 1.29. Since 1.123 < 1.29, the null hypothesis is not rejected, indicating insufficient evidence of unequal variances at the 5% level. The two-tailed p-value exceeds 0.05, further supporting non-rejection.^[19]

Role in ANOVA and Regression

The F-distribution is central to analysis of variance (ANOVA), where it serves as the sampling distribution for the test statistic used to compare means across multiple groups by partitioning the total observed variance into components attributable to differences between groups and within groups. In a one-way ANOVA, the between-group mean square (MSB) captures variability due to group differences, while the within-group mean square (MSW) reflects residual variability; the ratio F = \frac{\mathrm{MSB}}{\mathrm{MSW}} follows an F-distribution with k-1 numerator degrees of freedom (where k is the number of groups) and n-k denominator degrees of freedom (where n is the total sample size) under the null hypothesis of equal population means.^[23] This variance decomposition allows researchers to determine whether observed differences in group means are statistically significant beyond what would be expected by chance, with large F-values indicating potential treatment or factor effects.^[24] Ronald A. Fisher developed ANOVA and its reliance on variance ratios in the 1920s while working on agricultural experiments at Rothamsted Experimental Station, providing a framework to evaluate treatment efficacy in designed experiments by systematically allocating and analyzing variance sources. The associated distribution, later named the F-distribution by George W. Snedecor in 1934 to honor Fisher's contributions, formalized the probabilistic foundation for these tests.^[25] Power analysis for ANOVA F-tests typically employs effect size measures such as Cohen's f, with benchmarks of 0.10 for small, 0.25 for medium, and 0.40 for large effects, guiding sample size determination to achieve desired detection power (e.g., 0.80) against alternatives where group means differ.^[26] In multiple linear regression, the F-distribution underpins the overall significance test, which evaluates whether the model explains variance in the response variable better than an intercept-only model by comparing explained variance to unexplained residual variance. The test statistic is F = \frac{\mathrm{SSR}/p}{\mathrm{SSE}/(n-p-1)}, where SSR is the regression sum of squares, SSE is the error sum of squares, p is the number of predictors, and n is the sample size; under the null hypothesis H_0: \beta_1 = \cdots = \beta_p = 0, this follows an F-distribution with p and n-p-1 degrees of freedom.^[27] Valid inference from both ANOVA and regression F-tests requires assumptions of normally distributed errors, homoscedasticity (constant error variance), and independence of observations, violations of which can inflate Type I error rates or reduce test power.^[28] These methods extend to generalized linear models (GLMs), where Gaussian GLMs (equivalent to linear regression) directly employ F-tests, while non-Gaussian GLMs approximate F-tests using scaled deviance or leverage likelihood ratio tests for model comparisons under similar variance-stability assumptions.

References

[1]
1.3.6.6.5. F Distribution - Information Technology Laboratory
The F distribution is the ratio of two chi-square distributions, used for hypothesis tests and determining confidence intervals, like in analysis of variance.
[2]
4.2 - The F-Distribution | STAT 415
The F-distribution is a probability distribution used for the ratio of two variances, defined using independent chi-square random variables.
[3]
Stats: F-Test
The F-distribution is formed by the ratio of two independent chi-square variables divided by their respective degrees of freedom. Since F is formed by chi- ...
[4]
Introductory Econometrics Chapter 17: F Tests
The F-distribution is named after Ronald A. Fisher, a leading statistician of the first half of the twentieth century. This chapter demonstrates that the F ...
[5]
[PDF] A Summary of Some Connections Among Some ... - USC Dornsife
• F: the original idea can be traced back to Sir Ronald Aylmer Fisher (1890–1962), who, in early 1920-s introduced something similar, but in a slightly ...
[6]
Ronald Aylmer Fisher (1890-1962)
Fisher made important contributions to many areas of statistics, including but not limited to, study design, testing significance of regression coefficients.Missing: origin | Show results with:origin
[7]
10.1 - Introduction to the F Distribution | STAT 200
The F distribution is used in one-way ANOVAs to compute F test statistics, and has two degrees of freedom: between and within groups.Missing: definition | Show results with:definition
[8]
F-Distribution Tables - Statistics Online Computational Resource
May 28, 2020 · The F distribution is a right-skewed distribution used most commonly in Analysis of Variance. When referencing the F distribution, the ...
[9]
F-Distribution -- from Wolfram MathWorld
F-Distribution ; f_(n,m)(x), = (Gamma((n+m)/2)n^(n/ ; = (m^(m/2)n^(n/2)x ; F_(n,m)(x), = I((nx)/(m+nx);1/2n, ; = 2n^((n-2)/2)(x/m).Missing: derivation | Show results with:derivation<|control11|><|separator|>
[10]
F distribution | Properties, proofs, exercises - StatLect
A random variable X has an F distribution if it can be written as a ratio [eq1] between a Chi-square random variable $Y_{1}$ with $n_{1}$ degrees of freedom ...Definition · Relation to the Gamma... · Relation to the Chi-square... · Density plots
[11]
Proof: Probability density function of the F-distribution
Oct 12, 2021 · fX,Y(x,y)=fX(x)⋅fY(y). (8) With the probability density function of an invertible function, the joint density of F and W can be derived as: fF, ...
[12]
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05:_Special_Distributions/5.11:_The_F_Distribution
[13]
11.2 The F-Distribution – Introduction to Statistics
The F -distribution curve is skewed to the right, and its shape depends on the degrees of freedom. As the degrees of freedom increase, the curve of an F - ...
[14]
The F Distribution - Random Services
Recall that the beta function B can be written in terms of the gamma function by ... Moments. The random variable representation in definition [2], along ...
[15]
THE MOMENTS OF THE z AND F DISTRIBUTIONS | Biometrika
F. N. DAVID; THE MOMENTS OF THE z AND F DISTRIBUTIONS, Biometrika, Volume 36, Issue 3-4, 1 December 1949, Pages 394–403, https://doi.org/10.1093/biomet/36.
[16]
[PDF] mathematics: wilson and hilferty proc. nas
The possibility of obtaining so good an approximation to the complete integral, which is r(n/2) except for a multiplier, suggests that the distribution might be ...
[17]
[PDF] Distributions Derived From the Normal Distribution
Chi-Square Distributions. Definition. If Z ∼ N(0, 1) (Standard Normal r.v.) then. U = Z 2 ∼ χ1. 2 , has a Chi-Squared distribution with 1 degree of freedom.
[18]
13.2 The F Distribution and the F Ratio | Texas Gateway
The F distribution is derived from the Student's t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution.
[19]
1.3.5.9. F-Test for Equality of Two Variances
An F-test (Snedecor and Cochran, 1983) is used to test if the variances of two populations are equal. This test can be a two-tailed test or a one-tailed test.
[20]
12.2 - Two Variances | STAT 415
Let's now recall the theory necessary for developing a hypothesis test for testing the equality of two population variances.
[21]
1.3.5.10. Levene Test for Equality of Variances
Levene's test checks if k samples have equal variances, which is called homogeneity of variance. It verifies if the assumption of equal variances is valid.
[22]
Levene, H. (1960) Robust Tests for Equality of Variances. In Olkin, I ...
(1960) Robust Tests for Equality of Variances. In: Olkin, I., Ed., Contributions to Probability and Statistics, Stanford University Press, Palo Alto, 278-292.
[23]
2.6 - The Analysis of Variance (ANOVA) table and the F-test
The degrees of freedom associated with SSR will always be 1 for the simple linear regression model. The degrees of freedom associated with SSTO is n-1 = 49-1 = ...
[24]
How F-tests work in Analysis of Variance (ANOVA) - Statistics By Jim
An F-statistic is the ratio of two variances and it was named after Sir Ronald Fisher. Variances measure the dispersal of the data points around the mean.Missing: original motivation
[25]
6.11 - F distribution - biostatistics.letgen.org
Fisher first developed the concept of the variance ratio in the 1920s, calling it the “general z-distribution”. We can for illustration purposes define the F ...
[26]
[PDF] One-Way Analysis of Variance F-Tests using Effect Size - NCSS
This simplified procedure only requires the input of an effect size, usually f, as proposed by Cohen. (1988). Assumptions. Using the F test requires certain ...
[27]
6.2 - The General Linear F-Test | STAT 501
The "general linear F-test" involves three basic steps, namely: Define a larger full model. (By "larger," we mean one with more parameters.) ...
[28]
[PDF] Lecture 10: F-Tests, ANOVA and R2 1 ANOVA
Assumptions In deriving the F distribution, it is absolutely vital that all of the assumptions of the Gaussian-noise simple linear regression model hold: the ...