The short answer
A confidence interval is a range built around a sample statistic that's likely to contain the true population value: interval = point estimate ± (critical value × standard error). A 95% confidence interval means that if you repeated the sampling process many times, about 95% of the intervals you'd build would contain the true population parameter — not that there's a 95% chance the true value sits inside this one specific interval.
Key takeaways
- A wider confidence level (99% vs. 90%) always produces a wider interval — you're trading precision for certainty, not getting a "more correct" answer.
- Use the z-distribution when the population standard deviation is known or the sample is large (n ≥ 30); use the t-distribution when σ is estimated from a small sample.
- Quadrupling the sample size only halves the margin of error, since standard error shrinks with the square root of n, not with n itself.
- "95% confidence" describes the long-run reliability of the method across repeated sampling, not the probability that this specific interval contains the true value.
The confidence interval formula
Mean (σ known): CI = x̄ ± z × (σ / √n)
Proportion: CI = p̂ ± z × √(p̂(1 − p̂) / n)
| Confidence level | z critical value |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
z vs. t: which critical value to use
| Condition | Use | Why |
|---|---|---|
| σ known, any sample size | z | Standard normal distribution applies exactly |
| σ unknown, n ≥ 30 | z (approx.) | Large samples make t nearly identical to z |
| σ unknown, n < 30 | t | Fatter tails account for extra uncertainty in estimating σ |
Worked example: 95% interval for a sample mean
Sample mean x̄ = 50.5, sample size n = 100, standard deviation σ = 10.5, confidence level = 95%.
Standard error = σ / √n = 10.5 / √100 = 1.05
Margin of error = 1.96 × 1.05 ≈ 2.06
95% CI = 50.5 ± 2.06 = [48.44, 52.56]
We'd say we are 95% confident the true population mean falls between 48.44 and 52.56 — meaning this method captures the true mean in about 95% of samples drawn the same way, not that this specific interval has a 95% chance of being right.
Common mistakes to avoid
- Interpreting "95% confidence" as "95% probability the true value is in this interval" — it's a statement about the long-run method, not this one computed interval.
- Using z when the sample is small and σ is unknown — the t-distribution's fatter tails exist specifically to account for that extra uncertainty.
- Assuming doubling the sample size halves the margin of error — it actually takes quadrupling n to halve the margin, since standard error shrinks with √n.
- Forgetting proportion confidence intervals need p̂ between 0 and 1, and can behave poorly near 0 or 1 with small samples.
Related calculators
- Standard Deviation Calculator — compute σ from raw data before building an interval around it.
- Sample Size Calculator — work backward from a target margin of error to the sample size you need.
- Z-Score Calculator — check where an individual data point falls relative to this distribution.
- P-Value Calculator — test a specific hypothesis rather than estimating a range.