What does the F value tell you in ANOVA?

Answer 1

The purpose of a one-way ANOVA is to determine if the means of k populations are equal or not. We especially want to test the null hypothesis H₀: µ₁ = µ₂ = ... = µ_k against the alternative hypothesis H_a: all k means are not equal. When the k samples are randomly and independently selected from normal populations having equal population standard deviations (or, equivalently, equal population variances), and the null hypothesis is true, then the ratio of the treatment mean square to the error mean square has an F Distribution with degrees of freedom for the numerator df₁ = k - 1 and degrees of freedom for the denominator df₂ = n - k. This ratio, called the F Ratio, is the test statistic used to test the above hypotheses:

F equals Treatment Mean Square OR MSTR

Error Mean Square MSE

Note that MSTR equals Treatment Sum Of Squares OR SSTR .

k - 1 k - 1

Note that MSE equals Error Sum Of Squares OR SSE .

n - k n - k

SSTR = n₁(xbar₁ - xbar)² + n₂(xbar₂ - xbar)² + ... + n_k(xbar_k - xbar}²

SSE = (n₁ - 1)s₁² + (n₂ - 1) s₂² + ... + (n_k - 1)s_k²