Biostatistics Study NotesFrom concepts and derivations to interpretation

CHAPTER 04 · TOPIC 06

Inference for Population Variances

Use a chi-square test for one population variance and an F test for two variances, with links to pooled and Welch t tests.

On this page
  1. Chi-Square Test for One Population Variance
  2. Constructing the chi-square statistic
  3. Rejection region for two-tailed test
  4. F Test for Two Population Variances
  5. Why the F-test arises from two chi-square variables
  6. Relationship to the Pooled and Welch t Tests

Estimating the population variance determines whether the dispersion of the data is equal to a specified criterion or whether the variances of two populations are equal. In a normal population, the chi-square distribution is used for one population variance, and the F distribution is used for the ratio of two independent population variances.

Chi-Square Test for One Population Variance#

two-sidedH0:σ2=σ02,H1:σ2≠σ02H_0:\sigma^2=\sigma_0^2,\qquad H_1:\sigma^2\ne\sigma_0^2
left-tailedH0:σ2≥σ02,H1:σ2<σ02H_0:\sigma^2\geq\sigma_0^2,\qquad H_1:\sigma^2<\sigma_0^2
right-tailedH0:σ2≤σ02,H1:σ2>σ02H_0:\sigma^2\leq\sigma_0^2,\qquad H_1:\sigma^2>\sigma_0^2

Constructing the chi-square statistic#

If we estimate μ by X̄ in an independent sample of a normal population, we lose one degree of freedom, and the standardization of the sample variance follows a chi-square distribution with n−1 degrees of freedom.

μ known∑i=1n(Xi−μ)2σ2∼χn2\frac{\sum_{i=1}^{n}(X_i-\mu)^2}{\sigma^2}\sim\chi_n^2
Estimate μ with X̄(n−1)S2σ2∼χn−12\frac{(n-1)S^2}{\sigma^2}\sim\chi_{n-1}^2
Observation statistics under H₀χobs2=(n−1)S2σ02\chi_{\mathrm{obs}}^2=\frac{(n-1)S^2}{\sigma_0^2}

Rejection region for two-tailed test#

χobs2<χα/2, n−12orχobs2>χ1−α/2, n−12\chi_{\mathrm{obs}}^2<\chi_{\alpha/2,\,n-1}^2\quad\text{or}\quad\chi_{\mathrm{obs}}^2>\chi_{1-\alpha/2,\,n-1}^2

F Test for Two Population Variances#

Equal variance hypothesisH0:σ12=σ22,H1:σ12≠σ22H_0:\sigma_1^2=\sigma_2^2,\qquad H_1:\sigma_1^2\ne\sigma_2^2
F-statisticF=S12S22∼Fn1−1, n2−1F=\frac{S_1^2}{S_2^2}\sim F_{n_1-1,\,n_2-1}

If we exchange the numerator and denominator, F becomes 1/F, and we also exchange the two degrees of freedom.

Why the F-test arises from two chi-square variables#

Group 1U1=(n1−1)S12σ2∼χn1−12U_1=\frac{(n_1-1)S_1^2}{\sigma^2}\sim\chi_{n_1-1}^2
2nd groupU2=(n2−1)S22σ2∼χn2−12U_2=\frac{(n_2-1)S_2^2}{\sigma^2}\sim\chi_{n_2-1}^2
ratio divided by degrees of freedomU1/(n1−1)U2/(n2−1)=S12S22∼Fn1−1, n2−1\frac{U_1/(n_1-1)}{U_2/(n_2-1)}=\frac{S_1^2}{S_2^2}\sim F_{n_1-1,\,n_2-1}

Relationship to the Pooled and Welch t Tests#

MethodQuestionSetting
One-population variance χ² testσ²=σ₀²?one normal population
Two-population variance F testσ₁²=σ₂²?two independent normal populations
Pooled tμ₁=μ₂?Adds an equal-variance assumption
Welch tμ₁=μ₂?Does not require equal variances