Calculate confidence intervals for a mean instantly — no stats software needed
Sample Parameters
Distribution
Result
x̄ ± 3.9337 | t* = 2.0227, df = 39
Interval Visualisation
Formula
CI = x̄ ± t* × (s / √n)
= 68.5000 ± 2.0227 × (12.3000 / √40)
= 68.5000 ± 2.0227 × 1.9448
= 68.5000 ± 3.9337
= [64.5663, 72.4337]
Compare Across Confidence Levels
| Level | t* | Lower | Upper | ± ME |
|---|---|---|---|---|
| 80% | 1.304 | 65.965 | 71.035 | ±2.535 |
| 85% | 1.468 | 65.644 | 71.356 | ±2.856 |
| 90% | 1.685 | 65.223 | 71.777 | ±3.277 |
| 95% | 2.023 | 64.566 | 72.434 | ±3.934 |
| 99% | 2.708 | 63.234 | 73.766 | ±5.266 |
How It Works
Common Critical Values
| Level | z* | α/2 |
|---|---|---|
| 80% | 1.282 | 0.100 |
| 85% | 1.440 | 0.075 |
| 90% | 1.645 | 0.050 |
| 95% | 1.960 | 0.025 |
| 99% | 2.576 | 0.005 |
A confidence interval calculator estimates the range that likely contains the true population mean based on your sample data. Enter the sample mean, standard deviation, and sample size, then choose a confidence level (80%–99%). The calculator uses the t-distribution for small samples (recommended when population SD is unknown) or the Z-distribution for large or known-variance situations, computing the margin of error and exact interval bounds.
Use the t-distribution whenever the population standard deviation is unknown — which is almost always the case in practice. The t-distribution is slightly wider than Z, especially for small samples (n < 30), providing a more conservative and accurate interval. The Z-distribution is appropriate only when the population SD is known or the sample size is very large (n ≥ 200+).
A 95% CI means that if you repeated the sampling process many times and computed the interval each time, approximately 95% of those intervals would contain the true population mean. It does NOT mean there is a 95% probability that the true mean falls in this specific interval — the mean is a fixed (unknown) value, not a random variable.
Larger sample sizes produce narrower confidence intervals. The standard error (SE = s / √n) shrinks as n grows, which reduces the margin of error. Doubling your sample size narrows the interval by a factor of √2 ≈ 1.41. This is why precision comes at a steep cost: halving the interval width requires four times as many observations.
You need three values: (1) the sample mean (x̄), the average of your observations; (2) the sample standard deviation (s), how spread out the data is; and (3) the sample size (n), the count of observations. You also choose a confidence level — 95% is the most common in research — and whether to use t or Z critical values.