When do I use a chi-square test in CFA Level I, and how do I set up the hypothesis for variance testing?

For CFA Level I, the chi-square test is primarily tested in the context of hypothesis tests about a single population variance (or standard deviation). The goodness-of-fit application appears less frequently at Level I but is worth knowing conceptually.

Chi-Square Test for Variance:

The test statistic is:

χ² = (n - 1) × s² / σ₀²

Where:

• n = sample size
• s² = sample variance
• σ₀² = hypothesized population variance
• Degrees of freedom = n - 1

The chi-square distribution is right-skewed and non-negative, which means the critical value approach differs from the symmetric z-test or t-test.

Worked Example — Hargrove Fixed Income Fund

The compliance team at Hargrove Asset Management claims the monthly return volatility (standard deviation) of their bond fund is no more than 1.8%. A risk analyst collects 25 months of returns and calculates a sample standard deviation of 2.3%. Test at the 5% significance level.

Step 1 — Hypotheses:

• H₀: σ² ≤ (0.018)² = 0.000324
• H₁: σ² > 0.000324 (one-tailed, upper tail)

Step 2 — Test Statistic: χ² = (25 - 1) × (0.023)² / (0.018)² = 24 × 0.000529 / 0.000324 = 39.19

Step 3 — Critical Value: For α = 0.05, df = 24, the upper-tail critical value is χ²₀.₀₅ = 36.415

Step 4 — Decision: 39.19 > 36.415 → Reject H₀. There is sufficient evidence that the fund's volatility exceeds the claimed 1.8%.

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Important Properties:

1. The chi-square distribution is bounded below by zero
2. It is not symmetric — for two-tailed tests, you need both upper and lower critical values separately
3. As df increases, the distribution approaches normality
4. The test assumes the underlying population is normally distributed (sensitive to this assumption)

Exam Tip: When you see a question about testing whether a portfolio's risk exceeds or falls below a stated benchmark, think chi-square. Remember to square the standard deviations to get variances before plugging into the formula.

For additional hypothesis testing practice, check out our CFA Level I study materials.