📘 Chi-Square Distribution

🧭 When to Use the Chi-Square

• Testing **variance** of a normal population: \((n-1)S^2/\sigma_0^2 \sim \chi^2_{n-1}\).
• **Goodness-of-fit** tests (counts vs. expected), **independence** in contingency tables.
• As the distribution of a **sum of squared standard normals**: \(\sum_{i=1}^{k} Z_i^2\sim\chi^2_k\).
• Special case of **Gamma**: \(\chi^2_k \equiv \text{Gamma}(\alpha=k/2,\ \theta=2)\).

📐 Definitions & Key Formulas

Write \(X\sim\chi^2_k\) with degrees of freedom \(k>0\).

• PDF: \[ f(x)=\frac{1}{2^{k/2}\Gamma(k/2)}\,x^{k/2-1} e^{-x/2},\quad x>0 \]
• CDF: \(F(x)=P(X\le x)=\dfrac{\gamma(k/2,\ x/2)}{\Gamma(k/2)}\) (regularized lower incomplete gamma).
• Mean: \(\mathbb{E}[X]=k\)
• Variance: \(\mathrm{Var}(X)=2k\)
• Right-tail: \(P(X>x)=1-F(x)\). Common in hypothesis tests.

Excel mapping: CHISQ.DIST(x, df, TRUE) = CDF; CHISQ.DIST.RT(x, df) = right-tail \(P(X>x)\); CHISQ.INV(p, df) = lower-tail quantile; CHISQ.INV.RT(p, df) = right-tail critical value.

🛠️ Worked Example 1 — Right-Tail Probability

Scenario. Let \(X\sim\chi^2_{10}\). Find \(P(X>18.3)\).

Solution

We want the right tail: \(P(X>18.3)=1-F(18.3)\).

Excel: use the built-in right tail function:

=CHISQ.DIST.RT(18.3, 10)

Answer: \(P(X>18.3)\) ≈ Excel result (about 0.05–0.06)

Sanity check: For \(k=10\), mean \(=10\), SD \(=\sqrt{20}\approx 4.472\). Value 18.3 is roughly \((18.3-10)/4.472\approx 1.86\) SDs above mean → tail around a few percent.

🧰 Worked Example 2 — Critical Values for Tests

Scenario. We plan a **right-tailed** variance test with \(k=8\) degrees of freedom at significance \(\alpha=0.05\). Find the critical value \(c\) so that \(P(X>c)=0.05\) for \(X\sim\chi^2_{8}\). Also report the **central 95% interval** \([L,U]\) such that \(P(L\le X \le U)=0.95\).

Right-tailed cutoff

We need \(c=\chi^2_{8,\,0.05}\) (right-tail). In Excel:

=CHISQ.INV.RT(0.05, 8)

Answer: \(c\) = Excel value (≈ 15.507)

Central 95% interval

Lower 2.5% point \(L\): use lower-tail inverse with \(p=0.025\).

=CHISQ.INV(0.025, 8)

Upper 97.5% point \(U\): use right-tail inverse with \(0.025\).

=CHISQ.INV.RT(0.025, 8)

Answer: \([L,U]\) = (Excel values, ≈ 2.180, 17.535)

Use \([L,U]\) to form CI for a variance of a normal population: \(\displaystyle \left[\frac{(n-1)S^2}{U},\ \frac{(n-1)S^2}{L}\right]\).

🧾 Excel Quick Reference (Chi-Square)

• Right tail \(P(X>x)\): =CHISQ.DIST.RT(x, df)
• CDF \(P(X\le x)\): =CHISQ.DIST(x, df, TRUE)
• PDF \(f(x)\): =CHISQ.DIST(x, df, FALSE)
• Lower-tail quantile \(x_p\): =CHISQ.INV(p, df)
• Right-tail critical value \(c\) with \(P(X>c)=p\): =CHISQ.INV.RT(p, df)
• Goodness-of-fit / independence test: =CHISQ.TEST(observed_range, expected_range) (returns p-value)

🧪 Chi-Square Explorer (try it yourself)