The distributions and probability integrals of some quadratic forms applied in testing the null hypothesis (H₀) in the analysis of variance are reviewed. In the alternative hypothesis (H₁), the distributions of the quadratic forms that almost occur in evaluating the power function are investigated. The author gives in this paper: The cube-root normalization method of the non-central chi-square. Exact formulae for the distribution and probability integrals of a weighted sum of chi-squares with even degrees of freedom are given in finite series. The probability integral for a variable distributed as the ratio of two independent weighted sums of chi-squares, with even degrees of freedom, also given. Special cases and numerical examples are given for illustration and verification.