8.15
F Distribution
The F distribution was named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction) with two sets of degrees of fre…
The F test, named after the renowned statistician Sir Ronald Fisher, compares the difference between population variances of two normally distributed populations.
The F test uses the F statistic, which is the ratio of the sample variances and, thus, is never negative.
Generally, for ease of calculations, the numerator represents the higher sample variance while the denominator denotes the smaller sample variance.
As the difference between the sample variances reduces, the F statistic gets closer to unity.
Computing the F statistic for several random samples of two independent normally distributed populations, and plotting the F statistic yields the F distribution curve, an asymmetric curve, similar to the chi-square distribution curve.
However, unlike the chi-square-based tests, the F distribution has two sets of degrees of freedom, one for the numerator and another for the denominator. The exact shape of the F distribution curve depends on these two degrees of freedom.
This distribution is helpful in the F test and methods involving the comparison of variances, such as ANOVA.
View the full transcript and gain access to JoVE Core videos
Q1: What is the F statistic and why is it never negative?
The F statistic is a ratio of two sample variances from normally distributed populations. Because it represents the division of one variance by another, and variances are always non-negative values, the F statistic cannot be negative. By convention, the numerator contains the larger sample variance and the denominator contains the smaller variance, simplifying calculations.
Q2: How does the F distribution differ from the chi-square distribution?
Both the F distribution and chi-square distribution are asymmetric curves used in statistical testing. However, the F distribution has two sets of degrees of freedom—one for the numerator and one for the denominator—while the chi-square distribution has only one set. The shape of the F distribution depends on both sets of degrees of freedom, making it more flexible for comparing variances across multiple groups.
Q3: What are variance between samples and variance within samples in ANOVA?
Variance between samples estimates the population variance from the variance of sample means, representing variation due to treatment or explained variation. Variance within samples is the average of individual sample variances, representing variation due to error or unexplained variation. Together, these two variance estimates form the F ratio used in ANOVA to test whether group means differ significantly.
Q4: Why should ANOVA be used instead of multiple t tests when comparing more than two groups?
Performing multiple pairwise t tests increases the likelihood of making a Type 1 error, where you incorrectly reject a true null hypothesis. One-Way ANOVA expands the t test framework to compare more than two groups simultaneously, controlling the overall error rate. This single comprehensive test is more statistically efficient and reliable than conducting multiple individual comparisons.
Q5: How do degrees of freedom affect the shape of the F distribution?
The F distribution's exact shape depends on two sets of degrees of freedom: one for the numerator and one for the denominator. Different combinations of these degrees of freedom produce different distribution curves. As you compute the F statistic for multiple random samples from two independent normally distributed populations, the resulting F distribution curve reflects these specific degrees of freedom values.
Q6: What happens to the F statistic as the difference between sample variances decreases?
As the difference between sample variances reduces, the F statistic approaches unity, or a value of one. This occurs because the F statistic is calculated as the ratio of the larger sample variance to the smaller sample variance. When the two variances become more similar, their ratio approaches one, indicating less evidence of a significant difference between populations.
Q7: How is the F distribution derived from the Student's t distribution?
The F distribution is derived from the Student's t distribution, with the values of the F distribution being squares of the corresponding values of the t distribution. This mathematical relationship connects the two distributions, allowing the F test to extend hypothesis testing capabilities beyond what the t distribution alone can provide, particularly for comparing multiple group variances simultaneously.