Confidence Intervals: Concepts, Formulas and Examples

A confidence interval is a range of plausible values for an unknown population parameter, built from sample data: "we are 95% confident the true mean lies between 48 and 56." It upgrades a point estimate with an honest margin of error.

It matters because a single number hides how much you know. Reporting x̄ = 52 from 10 observations versus from 10,000 observations is a completely different claim, and the interval's width shows that difference. Confidence intervals are the standard language for reporting uncertainty in science, polls and quality reports.

Key Concepts

• Confidence level (1 - α): usually 95%. It means the method produces intervals that trap the true parameter 95% of the time over repeated sampling.
• Margin of error: the ± part, equal to (critical value) × (standard error). Wider with smaller n or higher confidence.
• Critical values: z* = 1.645 for 90%, 1.96 for 95%, 2.576 for 99%; use t* with n-1 degrees of freedom when σ is unknown and n is small.
• Interpretation: "95% confident" describes the method's long-run success rate, not the probability that this specific interval contains μ.
• Width trade-off: higher confidence or smaller samples give wider intervals; more data is the only way to get both narrow and confident.

Formulas

Mean, σ known or large n: x̄ ± z* × σ/√n. Mean, σ unknown, small n: x̄ ± t* × s/√n with n-1 degrees of freedom. Proportion: p̂ ± z* × √(p̂(1-p̂)/n), valid when np̂ ≥ 10 and n(1-p̂) ≥ 10. Assumptions: random sample, independent observations; the t-interval also wants roughly normal data when n is small. A 95% interval is not "95% of the data lie inside", it is a statement about the parameter.

Worked Example

In a survey of n = 100 voters, 60 support a policy, so p̂ = 0.60. Build a 95% confidence interval. Step 1: SE = √(0.6 × 0.4/100) = √(0.0024) ≈ 0.0490. Step 2: z* = 1.96. Step 3: margin = 1.96 × 0.0490 ≈ 0.0960. Step 4: interval = 0.60 ± 0.096 = (0.504, 0.696). We are 95% confident the true support lies between 50.4% and 69.6%. Note the interval stays above 50%, so a claim of majority support is consistent with the data, though only just.

Common Misconceptions

• It is not a probability about μ: after computing (0.504, 0.696), μ either is or is not inside; the 95% refers to the procedure's long-run hit rate, not this interval's chance.
• Narrower is not always better: a narrow interval from a biased sample or wrong formula is precisely wrong. Validity comes before width.
• Overlapping intervals do not prove "no difference": two 95% intervals can overlap while a proper test still finds a significant difference. Compare with the right test, not by eye.

Real-World Applications

• Election polls: "52% ± 3%" is a 95% confidence interval for the true vote share.
• Medicine: a trial reports the drug's effect as an interval so doctors see both the estimate and its precision.
• Manufacturing: an interval for the mean fill weight decides whether machines need recalibration or the variation is just sampling noise.

A confidence interval wraps a point estimate in its uncertainty: estimate ± critical value × standard error, with z* or t* chosen for the confidence level. Read it as the method's track record, not a probability about the parameter, and let the width tell you how much the data really say.