Session 8.3 — CI for Variance (χ²) and Standard Deviation

Why a CI for variance?

Sometimes the key question is the variability, not just the mean—for example, process spread in QC, calibration uncertainty, or risk buffers. Under Normality, the sample variance \( S^2 \) links to a chi-square:

\[ \frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{\nu}, \quad \nu=n-1. \] A two-sided \(100(1-\alpha)\%\) CI for \( \sigma^2 \) can be written with left-tail χ² quantiles as \[ \boxed{ \frac{(n-1)S^2}{\chi^2_{\,1-\alpha/2,\nu}} \le \sigma^2 \le \frac{(n-1)S^2}{\chi^2_{\,\alpha/2,\nu}} } \] (Here \( \chi^2_{p,\nu} \) means the value with CDF\(=p\).) For \( \sigma \), take square roots of the endpoints.

Interpretation: if we repeat the whole process many times, about \(100(1-\alpha)\%\) of those intervals would cover the true variance (or SD).

Worked example (raw data or summary)

Enter raw data to compute \( s \) and \( n \), or use the summary boxes directly. Choose confidence and compute.

Presets:

Enter raw data (comma/space/newline separated)

Sample SD \( s \)

Sample size \( n \)

Confidence

CI for variance \( \sigma^2 \)

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CI for standard deviation \( \sigma \)

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df \(=\nu=n-1\)

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χ² cutoffs (left-tail CDF values)

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Graph: \( \sigma \) CI whisker (not variance). Scale auto-adjusts.

Excel — quick steps (two-sided CI)

Inputs: \( s \) in B2, \( n \) in B3, ALPHA in B4 (e.g., 0.05 for 95%).
df: =B3-1
Left-tail χ² quantiles (preferred):
- \( \chi^2_{\alpha/2,\nu} \): =CHISQ.INV(B4/2, B5)
- \( \chi^2_{1-\alpha/2,\nu} \): =CHISQ.INV(1-B4/2, B5)
Right-tail alternative (equivalent): CHISQ.INV.RT(1-B4/2,df) and CHISQ.INV.RT(B4/2,df).
Variance bounds:
- Lower \( \sigma^2_L \): =(B5*B2^2)/CHISQ.INV(1-B4/2,B5)
- Upper \( \sigma^2_U \): =(B5*B2^2)/CHISQ.INV(B4/2,B5)
SD bounds: =SQRT(σ²_L) and =SQRT(σ²_U)

Assumption: The χ² CI relies on approximate Normality of the data. It is not robust to heavy skew/outliers with very small \( n \).

What changes the width?

Higher confidence (99% vs 95%) → wider CI.
Larger \( n \) → narrower CI.
More variability \( s \) → wider CI.

Tips for students

Report both: CI for \( \sigma^2 \) and for \( \sigma \) (square roots of bounds).
Show df and the two χ² cutoffs you used.
If Normality is doubtful at small \( n \), mention it and consider alternatives (e.g., bootstrapping \( s \)).

How to use t and χ² CIs in practice

Rule of thumb: use a t-interval when your question is about the mean (σ unknown). Use a χ² interval when your question is about the variability (σ or σ²).

Student Life — Commute time

Goal A (mean): “Is average one-way commute ≤ 30 minutes?” Sample: \( \bar x=31.4 \) min, \( s=8.0 \) min, \( n=10 \), 95%.

t-CI for mean (df=9):

\( \text{SE} = \frac{8}{\sqrt{10}} \approx 2.530,\quad t_{0.025,9}\approx 2.262 \)

\( \text{ME} = 2.262\times 2.530 \approx 5.72 \)

\( \mu \in [31.4-5.72,\ 31.4+5.72] = [25.68,\ 37.12] \) minutes

Decision: 30 is inside the interval → we can’t claim the mean is definitively ≤ 30. Gather more data or accept uncertainty.

Goal B (variability): “Is commute spread stable (σ ≤ 6 min)?” Same sample \( s=8.0,\ n=10 \), 95%.

χ²-CI for σ (df=9; using left-tail χ²): \( \chi^2_{0.025,9}\approx 2.700,\ \chi^2_{0.975,9}\approx 19.02 \)

Variance CI: \( \big[\,\frac{9\cdot 8^2}{19.02},\ \frac{9\cdot 8^2}{2.700}\big] \approx [30.3,\ 213.3] \)

σ-CI: \( [\sqrt{30.3},\ \sqrt{213.3}] \approx [5.5,\ 14.6] \) minutes

Decision: Since the σ-CI includes values > 6, we cannot claim the commute variability meets the ≤6-minute stability goal. Try larger \( n \) or interventions that reduce variance.

Engineering QC — Machined diameter

Spec: Target 42.00 mm, tolerance ±0.10 mm; desire process σ ≤ 0.10 mm. Sample: \( \bar x=41.95 \) mm, \( s=0.14 \) mm, \( n=20 \), 95%.

t-CI for mean (df=19): \( \text{SE} = \frac{0.14}{\sqrt{20}} \approx 0.0313,\ t_{0.025,19}\approx 2.093 \)

\( \text{ME} \approx 2.093 \times 0.0313 \approx 0.0655 \)

\( \mu \in [41.8845,\ 42.0155] \) mm

Interpretation: Mean is close to target; the CI barely dips below 41.90 → watch for slight under-sizing.

χ²-CI for σ (df=19): \( \chi^2_{0.025,19}\approx 8.907,\ \chi^2_{0.975,19}\approx 32.852 \)

Var CI: \( \big[\,\frac{19\cdot 0.14^2}{32.852},\ \frac{19\cdot 0.14^2}{8.907}\big] \approx [0.0113,\ 0.0418] \)

σ-CI: \( [0.106,\ 0.205] \) mm

Decision: Even the best-case bound (~0.106) exceeds 0.10 → process variability likely fails spec. Reduce sources of variation (tool wear, fixturing, calibration) before ramping.

Choosing the right interval

Question about a mean? Use t-interval (σ unknown). Show \( \bar x, s, n, \text{df}, t \), CI, and your decision in words tied to the practical threshold.
Question about variability (risk/spread)? Use χ²-interval for \( \sigma^2 \) or \( \sigma \). Show df and the two χ² cutoffs you used.
Both matter (common in QC & SLAs): compute both CIs and make a joint decision (center and spread).

Biases & pitfalls (and how to handle them)

Sampling bias: Only morning commutes? Only one operator/shift? Fix: randomize times, use stratified sampling (rush vs off-peak; operator A/B/C), ensure coverage.
Non-independence: Back-to-back parts or same route/day cause autocorrelation. Fix: sample across days/batches; space out runs; mix machines/fixtures.
Non-Normal data: χ² CI assumes Normality for σ. Heavy skew/outliers (e.g., accidents) inflate \( s \). Fix: larger \( n \); consider transforming (log-minutes) or bootstrapping σ for a robustness check (state this in your write-up).
Measurement error: Phone GPS lag; calipers out of calibration. Fix: calibrate tools; repeat-measure a standard; average repeated reads.

Underpowered studies: Wide CIs lead to “no decision.” Fix: increase \( n \) using your margin-planning formulas (z first, then refine with t).
Cherry-picking: Don’t drop “bad days” or “bad parts.” Fix: pre-register your inclusion rules; report sensitivity with/without outliers.
Shifting process: Tool wear or seasonal traffic changes mean data aren’t i.i.d. Fix: segment by time; monitor drift; re-estimate regularly.

Career connections

Operations / SLAs: Use t-CIs to show customers “95% CI for average response < target,” and χ²-CIs to guarantee stability of response-time variance.
Manufacturing QA: Pair t-CI (centering) with χ²-CI (spread) before releasing lots; tie σ-CI to capability (Cp/Cpk) narratives.
Healthcare / Service: Wait-time improvement pilots: t-CI for mean wait; χ²-CI to ensure fewer extreme delays.
Finance / Risk: t-CI for average P&L uplift; χ²-CI for volatility bands when presenting strategies to PMs/committees.

Student reporting checklist

Question & threshold: State the business/engineering question in words.
Inputs: \( \bar x, s, n \) (or raw data path), chosen confidence, assumptions.
Show work: df, critical values (t or χ²), SE/ME, and the CI with units.
Decision in words: Tie CI to the threshold (e.g., “cannot claim ≤ 30 min”).
Bias check: Note sampling and Normality issues; propose one improvement.

Session 8.3 — Confidence Interval for Variance (χ²)