Confidence Intervals Explained — What They Mean and How to Calculate Them

What a Confidence Interval Really Means

A 95% CI does NOT mean "there is a 95% probability that the true parameter lies in this interval." The true parameter is fixed (not random) — it either is or is not in the interval. The probability refers to the procedure, not this specific interval.

The correct interpretation: "We used a procedure that produces intervals covering the true parameter 95% of the time. This particular interval was produced by that procedure." If you built 100 such intervals from 100 different samples, about 95 would contain the true value and 5 would miss it.

In practice, most people use a shorthand: "We are 95% confident the true value lies between [lower] and [upper]." This is an acceptable simplification as long as you understand the frequentist meaning behind it.

Critical Values for Common Confidence Levels

The critical value (z* or t*) depends on your chosen confidence level and whether you know the population SD (σ). Use z* when σ is known or n is very large; use t* (with df = n−1) when σ is unknown and n is moderate.

Worked Example — Mean Delivery Time

A courier company samples 50 deliveries. Sample mean: x̄ = 4.2 days. Sample SD: s = 1.1 days. Build a 95% CI.

Step 1 — Standard error: SE = s / √n = 1.1 / √50 = 1.1 / 7.07 = 0.156 days.

Step 2 — Degrees of freedom: df = n − 1 = 49. t* ≈ 2.010 (at 95%, df=49).

Step 3 — Margin of error: ME = t* × SE = 2.010 × 0.156 = 0.313 days.

Step 4 — CI: [4.2 − 0.313, 4.2 + 0.313] = [3.887, 4.513] days.

Interpretation: We are 95% confident the true mean delivery time for all deliveries is between 3.9 and 4.5 days. If the company's target is 4.0 days, 4.0 is within the CI — we cannot reject the claim that the mean equals 4.0 days.

Confidence Interval for a Proportion

For a sample proportion p̂, the 95% CI is: p̂ ± z* × √(p̂(1−p̂)/n).

Example: In a survey of 400 voters, 220 (55%) support Candidate A. Build a 95% CI: p̂ = 0.55, SE = √(0.55 × 0.45 / 400) = √(0.000619) = 0.0249. CI: 0.55 ± 1.96 × 0.0249 = [0.501, 0.599].

Interpretation: We are 95% confident the true proportion supporting Candidate A is between 50.1% and 59.9%. The margin of error is ±4.9 percentage points.

Conditions for validity: np̂ ≥ 5 and n(1−p̂) ≥ 5. Here: 400 × 0.55 = 220 ≥ 5 ✓. For smaller p̂ or n, use the Wilson score interval or exact binomial interval instead.

What Affects Confidence Interval Width

Wider intervals mean less precise estimates. Three factors control CI width:

1. Sample size: Larger n → smaller SE → narrower CI. Doubling n reduces CI width by √2 ≈ 1.41 (41% narrower).

2. Confidence level: Higher confidence (99% vs 95%) requires a larger critical value → wider CI. You are trading precision for certainty.

3. Population variability: Larger σ or s → wider CI. You cannot control this without better measurement tools or homogeneous sampling.

Confidence Intervals vs P-Values

Both CI and p-values address the same underlying question, but from different angles. For a two-tailed hypothesis test at level α, the null hypothesis is rejected if and only if the (1−α)×100% CI does not contain the null value.

However, CIs give additional information: they show the size of the effect and whether it is practically meaningful. A p-value of 0.001 tells you an effect is statistically significant; a CI of [0.001, 0.003] units tells you it is tiny in practice. A CI of [5, 50] units tells you it is large and practically important.

Modern statistical practice increasingly prefers reporting CIs over bare p-values. Many journals now require CIs alongside p-values, and some fields (psychology, medicine) are moving toward CI-centric reporting.