A single point estimate is useful but silent about its own reliability; a confidence interval attaches an honest range. The general recipe, used identically across every version below:
point estimate ± (critical value) × (standard error)What does a 95% confidence level mean?
What the confidence level actually means: get this exactly right. A 95% confidence level does not mean "there is a 95% probability the true parameter lies in this particular interval." The true parameter is a fixed, unknown number, it either is or is not inside the one specific interval you just computed, full stop, and no probability attaches to that fact once the interval is built. What the 95% describes is the procedure: if you repeated the entire sampling-and-interval-construction process many times, about 95% of the resulting intervals (not this one specifically, all of them collectively) would contain the true parameter. This is the single most misquoted idea in applied statistics, finance included, and it is Common Mistakes item 1 for exactly that reason.
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How do you calculate a confidence interval for a mean?
Case 1: mean, σ known (z-based). Rare in practice (you almost never actually know the population σ) but the cleanest starting case, and the direct foundation for every case after it:
x̄ ± z* × (σ / √n)using the standard z-critical-values 1.6449 (90% confidence), 1.9600 (95%), and 2.5758 (99%), all verified directly, and all following from the empirical-rule logic.
Case 2: mean, σ unknown (t-based). The realistic case, and the one that matters most: replace the unknown σ with the sample standard deviation s, and replace the normal critical value with a t critical value at df = n − 1 degrees of freedom, to honor the extra estimation uncertainty:
x̄ ± t*(df = n − 1) × (s / √n)Worked Example 6 builds this in full on a small sample of IPO first-day returns.
Confidence interval for a proportion or a variance
Case 3, a proportion. Because a sample proportion p̂ is itself (up to a scale factor) a binomial count, and the CLT applies to it the same way it applies to a mean, a confidence interval for a population proportion p uses the normal approximation with its own standard-error formula:
p̂ ± z* × √[ p̂(1 − p̂) / n ]Practice Problem 16 drills this exactly.
Case 4, a variance. Using the chi-square fact from the distribution catalog, (n − 1)s²/σ² follows a chi-square distribution with n − 1 degrees of freedom, a confidence interval for the population variance σ² is:
[ (n − 1)s² / χ²_U , (n − 1)s² / χ²_L ]where χ²_U is the upper chi-square critical value (the value with (1 − α/2) of the distribution below it) and χ²_L is the lower one (with only α/2 below it), notice the larger critical value produces the smaller bound and vice versa, because it sits in the denominator. Worked Example 7 works this fully.
What makes a confidence interval wider or narrower?
What moves a confidence interval's width, three levers, memorize the direction of each. A larger sample size n narrows the interval (standard error shrinks as 1/√n). A higher confidence level widens it (you need a bigger critical value to be more sure you've captured the truth). More underlying variability (a bigger σ or s) widens it (a noisier population makes any fixed-size sample less informative). Every confidence interval in this module's Worked Examples and Practice Set is a specific instance of one of these four cases, moved by exactly these three levers.
The explicit hand-off to QM1.02. This module deliberately stops here, at estimation, and goes no further, not because the next step isn't interesting, but because teaching it here would duplicate a sibling module's job (R1). QM1.02 (Hypothesis Testing, Correlation & Regression) is where a confidence interval's logic is turned into a formal decision procedure: stating a null and alternative hypothesis, choosing a significance level, computing z, t, chi-square, and F test statistics and their p-values, weighing Type I error against Type II error and statistical power, testing whether a correlation is genuinely different from zero, and building simple and multiple regression models with their full diagnostic panel (heteroskedasticity, serial correlation, multicollinearity, misspecification). Every distribution this module introduced, normal, t, chi-square, F, reappears there as a test statistic's distribution rather than an estimator's distribution; the shapes are identical, the use is different, and QM1.02 is where that use is taught. QM1.03 (Time-Series, Monte Carlo & Intro ML-for-Finance) picks up the thread again for trend/AR/MA models, stationarity, and Monte Carlo construction, before handing every actual build off to the quant-code region in turn.