How to use the CHISQ.INV.RT() function in Excel

This function returns the inverse of the right-tailed chi-square (χ²) distribution, providing the critical value where:

• If probability = CHISQ.DIST.RT(x, df), then CHISQ.INV.RT(probability, df) = x.

Syntax

CHISQ.INV.RT(probability; degrees_freedom)

Purpose

Used in hypothesis testing to:

1. Determine critical values for χ² tests (e.g., goodness-of-fit, independence).
2. Validate whether observed results significantly deviate from expected results under the null hypothesis.

Arguments

Background

1. Right-Tailed χ² Distribution:
   • Models the sum of squared deviations from expected values.
   • Used when testing « greater than » hypotheses (e.g., variance exceeds a threshold).
2. Key Concepts:
   • Critical Value (x):
     • The χ² value beyond which the null hypothesis is rejected.
     • Calculated as CHISQ.INV.RT(α, df).
   • Degrees of Freedom (df):
     • Depends on the test type. For a 2×2 contingency table, df = 1.
3. Inverse Relationship:
   • CHISQ.INV.RT(α, df) is the inverse of CHISQ.DIST.RT(x, df).

Example: Vitamin C Efficacy Study

Scenario

• Goal: Test if Vitamin C reduces cold risk (null hypothesis: no effect).
• Data:
  • Expected cold cases (no Vitamin C): 22/936.
  • Observed cold cases (Vitamin C group): Fewer than 22.
• Significance Level (α): 2.5% (one-tailed test).

Step 1: Calculate Critical Value

CHISQ.INV.RT(0.025; 1)  // Returns 5.0239

• Interpretation: If the test statistic (v) > 5.0239, reject the null hypothesis.

Step 2: Compute Test Statistic (v)

1. For each category, calculate:

1. • Oi​ = Observed frequency.
   • Ei​ = Expected frequency.
2. Result: Suppose v = 6.47 (see Figure below).

Step 3: Compare v to Critical Value

• 6.47 > 5.0239 → Reject the null hypothesis.
• Conclusion: Insufficient evidence to confirm Vitamin C reduces colds at α = 2.5%.

Key Notes

1. When to Use:
   • Goodness-of-Fit Tests: Compare observed vs. expected frequencies.
   • Independence Tests: Check if two categorical variables are related.
2. Degrees of Freedom:
   • For a contingency table: df = (rows – 1) * (columns – 1).
   • For variance tests: df = sample size – 1.
3. Common Errors:
   • #NUM! if:
     • probability ≤ 0 or ≥ 1.
     • degrees_freedom < 1.