Testing variances: chi-square and F

In short

• One variance, normal population: $\chi^2 = (n-1)s^2/\sigma_0^2$ with $n - 1$ df.
• Chi-square is never negative and is skewed to the right, so its lower and upper critical values are not mirror images.
• Two variances, independent samples from normal populations: $F = s_1^2/s_2^2$ with $n_1 - 1$ (numerator) and $n_2 - 1$ (denominator) df.
• F is also never negative. An F near 1 means similar variances.
• Always work in variances: square every standard deviation first, including the hypothesised one.