Confidence Interval Calculator
Input Sample Parameters
Results
| Confidence Interval | — |
|---|---|
| Margin of Error (ME) | — |
| Critical Value (z* or t*) | — |
| Standard Error of Mean (SE) | — |
| Degrees of Freedom (df) | — |
A single sample mean is never the whole story—you measured 40 respondents, not the entire city. Confidence intervals turn that average into a range that accounts for sample size and spread, which is what stakeholders actually want before they green-light a decision.
How to Use This Confidence Interval Calculator
- Pick Z-interval or T-interval. Choose population SD (σ) known for a Z-interval, or sample SD (s) unknown for a T-interval—the default most field studies need.
- Enter sample mean (x̄), standard deviation, and sample size (n). SD must be positive; n must be a whole number (at least 2 for T-interval).
- Select confidence level. Presets include 90%, 95%, and 99%, or enter a custom level between 0% and 100%.
- Press Calculate. Results show the interval [lower, upper], margin of error, critical value, standard error, and degrees of freedom. Step-by-step math appears below the table.
Confidence Interval Formulas and Practical Applications
Think of the interval as a net cast around your sample mean: wider nets (lower confidence or smaller n) catch the true mean more often; tighter nets trade certainty for precision.
Standard error and margin of error
SE = s or σ / √n
ME = Critical Value × SE
Confidence Interval = x̄ ± ME
SE shrinks when n grows—doubling sample size cuts SE by about √2, which is why larger surveys produce narrower ranges without changing confidence level.
Z-interval vs. T-interval
Z-interval uses the normal critical value z* when σ is known (rare outside textbook problems). T-interval uses t* with df = n − 1 when you estimate spread from the sample—typical for polls and lab replicates. The t curve has heavier tails, so intervals widen slightly versus Z at the same confidence.
Worked intuition
Sample mean 50, s = 8, n = 30, 95% confidence: SE ≈ 8 / √30 ≈ 1.46. Multiply by t* ≈ 2.045 for df = 29 to get ME, then add and subtract from 50. Planning n upfront? Pair this with the Sample Size Calculator.
Frequently Asked Questions
What is a confidence interval?
It is a range around x̄ that estimates where the population mean likely falls, at the confidence level you select. The results table prints explicit lower and upper bounds.
When should I use a Z-interval vs. a T-interval?
Z when σ is genuinely known; T when you plug in sample s. If unsure, T is the safer real-world choice.
Why must sample size be at least 2 for a T-interval?
T needs df = n − 1 ≥ 1. One observation cannot estimate sample spread, so the tool rejects n < 2 for T-interval mode.
What is margin of error?
ME is half the interval width: critical value times SE. It is the plus-or-minus figure quoted in headlines ("± 3 points").
Does a 95% confidence interval mean 95% of data fall inside?
No—it describes uncertainty about the mean, not individual values. For Z-scores and tail areas, see the Z-score Calculator or browse math calculators.