Confidence Interval Calculator
Estimate the range in which the population mean is likely to fall from a sample — automatically uses the t-distribution for small samples.
Confidence intervals explained, step by step
Each concept below has a plain-English definition, its formula, a small diagram and a worked example — all in one card so you never have to jump between sections to piece it together.
1. What a confidence interval is
A confidence interval is a range of plausible values for a population parameter built from sample data. A 95% CI means the procedure — over many repeated samples — would trap the true mean about 95% of the time.
2. The margin of error formula
Margin of error is the critical value times the standard error. The standard error shrinks with the square root of the sample size, so quadrupling n only halves the width of the interval.
3. Z vs Student t
Use Z when σ is known or n is large (≥ 30). Use t when you estimate σ from a small sample — the heavier t-tails widen the interval to account for the extra uncertainty. As df grows, t converges to Z.
4. Choosing a confidence level
Higher confidence means a wider interval — more certainty about coverage comes at the cost of precision. 95% is the default in most sciences; use 90% for exploratory work, 99% for high-stakes decisions.
Z-values for common confidence levels
| Confidence level | Z-value (two-tailed) |
|---|---|
| 70% | 1.036 |
| 75% | 1.150 |
| 80% | 1.282 |
| 85% | 1.440 |
| 90% | 1.645 |
| 95% | 1.960 |
| 98% | 2.326 |
| 99% | 2.576 |
| 99.5% | 2.807 |
| 99.9% | 3.291 |
Features of this calculator
- Auto-switches to the Student t-distribution when n < 30 — a genuine correctness upgrade over Z-only tools.
- Explicitly labels which method was used and, for t, the degrees of freedom.
- Interval shown in all three common formats at once: x̄ ± MoE, ±%, and [lower – upper].
- Compact error-bar visual with the mean marker and both bounds labeled.
- Live-computed Z and t critical values for any confidence level between 80% and 99.9%.
- Step-by-step working shows the critical value, standard error, margin of error and final interval.
Frequently asked questions
+What is the difference between confidence level and confidence interval?
The confidence level is a number you choose (e.g. 95%) that controls the procedure's long-run coverage. The confidence interval is the range of values you get from a specific sample using that procedure.
+How do I make my confidence interval narrower?
Collect more data (MoE shrinks with √n), reduce variability in the measurement, or lower the confidence level — but a lower confidence level means the procedure covers the true mean less often over many repeats.
+Why does the tool sometimes show a t critical value bigger than the Z value?
Because the t-distribution has heavier tails. For df = 10 at 95%, t = 2.228 vs Z = 1.960 — the wider interval correctly reflects the extra uncertainty in estimating σ from a small sample.
+Does this assume the data is normally distributed?
The interval for the mean is exact if the underlying data is normal. For non-normal data with a reasonably large sample (n ≥ 30), the Central Limit Theorem makes the Z interval a good approximation for the mean.
+Can I use this for proportions?
This tool is for the mean of a continuous variable. For a proportion, use the Sample Size Calculator's margin-of-error mode, which uses p(1−p) as the variance term.