Intermediate 13 min read Statistics

Confidence Intervals

Point estimates need ranges to be honest

A confidence interval is a range that captures the population parameter with a chosen confidence, e.g., 95% of such intervals. It converts a point estimate into a statement with honest uncertainty.

confidence intervalmargin of error95%point estimateuncertainty

Prerequisites: Sampling and Inference

Before we start

What you will learn

Interpret a confidence interval correctly
Construct CIs for means and proportions
Explain why width reflects uncertainty, not bad luck

Introduction

What is this?

A 95% confidence interval for a parameter is the range μ̂ ± 1.96·SE: constructed this way, 95% of such intervals across repeated samples would contain the true parameter. It is NOT '95% chance the true value is in this specific interval' — the parameter is fixed; the procedure is what's probabilistic. Wider CIs mean less certainty; bigger samples narrow them.

Motivation

Why do you need it?

A point estimate alone invites false precision — 'revenue is 4.3% flat' hides a wide band.
CIs make decisions reasonable: 'the lift could be -1% to +4%' changes the bet.
They encode sample size and variance transparently.
They pair with tests: a 95% CI excluding 0 ≈ significance at α=0.05.

Think about it

Mental model

Imagine

Betting on a target with a dartboard's halo

Think of

Confidence interval

The dart (estimate) may be off, but the halo (interval) says how far off it might plausibly be. A small halo = confident shot; a huge halo = urgent need for more darts (data).

Mechanics

How it works

Compute the point estimate and its standard error SE.
Choose the confidence level; the z-multiplier (1.96 for 95%, 1.645 for 90%).
Interval = estimate ± z·SE.
Interpretation: in 95 of 100 repeated experiments, such intervals capture the truth.
Narrow it with a bigger sample or lower variance; only tighten the level by accepting more risk.

Repeated sampling: 95% of CIs capture the truth.

Sampleestimate ± SE

Multiply by z1.96 for 95%

IntervalBracket around estimate

Repeat~95% bracket the parameter

Hands-on

Examples

CI for a mean in one line

import numpy as np
np.random.seed(5)
x = np.random.normal(100, 12, 150)
n, mean, s = len(x), x.mean(), x.std(ddof=1)
se = s/np.sqrt(n)
lo, hi = mean - 1.96*se, mean + 1.96*se
print(f"mean={mean:.2f} 95% CI [{lo:.2f}, {hi:.2f}]")

The 95% CI brackets the sample mean by 1.96 standard errors — the analog of 'estimate ± uncertainty'.

Real world

In production

Executive dashboards show revenue change as a CI, not a single 'up/down' arrow.
Survey reporting prints margin of error (half-width) with every CSAT score.
Forecast baselines ship with 80%/95% bands for capacity planning.

Guidelines

Best practices

Always report intervals for estimates that will be compared or budgeted against.
Prefer 90–95%; avoid absurd 99.9% bands that look like 'no accuracy at all'.
For skewed metrics, consider bootstrap CIs instead of the normal approximation.
State n and variance alongside the interval so readers see what drives width.

Watch out

Common mistakes

Saying 'there's a 95% chance the truth is in THIS interval' (the parameter isn't random).
Reading non-overlapping CIs as 'proven different' without a proper test.
Using the population SD when only the sample is known.
Overstating precision for log-normal-like data by using the normal-approx interval.

When things break

Debugging tips

Absurdly narrow CI? Check you divided by n instead of sqrt(n), or used population σ.
CI wider than the data range? Likely a typo'd scaling or heavy outliers.
Interval hugs zero but test p ≈ 0.05 — report the interval and its implications, not the cutoff.
Bootstrap CI wildly different from normal? Returns skewed data — trust the bootstrap.

Interviews

Interview checkpoint

Test yourself

Quick quiz

0/2

A 95% CI for a mean is computed as estimate ±:

Doubling the sample size roughly halves the CI width.

Go deeper

Official documentation

NIST — confidence intervals

docs