Confidence Interval Calculator
Build confidence intervals for a mean or proportion at any confidence level.
About Confidence Interval Calculator
A confidence interval gives a range of plausible values for an unknown population parameter. A 95 % interval means: if you repeated the study many times, 95 % of the constructed intervals would contain the true parameter. It does NOT mean the parameter has a 95 % chance of being in this specific interval. When σ is known you use a z critical value; when σ is estimated from the sample you use a t critical value (with n−1 degrees of freedom), which produces a wider interval that accounts for the added uncertainty.
Formulas
Mean (σ known)
x̄ ± z* × σ/√n
- z*
- Critical z value
- σ
- Population std deviation
- n
- Sample size
Mean (σ unknown)
x̄ ± t* × s/√n, with df = n−1
Proportion
p̂ ± z* × √(p̂(1−p̂)/n)
Worked example
With the default values below, Confidence Interval Calculator returns Lower bound of 46.4216. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Interval typeone of Mean — σ known (z-interval), Mean — σ unknown (t-interval), Proportion (z-interval)
- Mean — σ known (z-interval)
- Sample mean (x̄)
- 50
- Population std deviation (σ)minimum 0
- 10
- Sample std deviation (s)minimum 0
- 10
- Sample size (n)minimum 2
- 30
- Sample proportion (p̂)accepts 0–1
- 0.6
- Sample size (n)minimum 1
- 100
- Confidence level (%)accepts 50–99.99
- 95
Results
- Lower bound
- 46.4216
- Upper bound
- 53.5784
- Margin of error
- 3.5784
- Critical value (z* or t*)
- 1.9600
- Standard error
- 1.825742
Step by step
- 1Critical z value= 1.9600
- 2Standard error= 1.8257
- 3Margin of error= 3.5784
- 4Confidence interval= [46.4216, 53.5784]
Assumptions and limits
- The result is derived only from the 8 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does a 95 % confidence interval actually mean?
It means the procedure used to construct the interval would capture the true parameter 95 % of the time across repeated samples. For this particular interval, the parameter is either inside it or it isn't — there is no probability attached to that fact.
When should I use t instead of z?
Always use t when the population standard deviation σ is unknown and you are estimating it from the sample. The t-distribution has heavier tails than the normal, producing a wider interval that honestly reflects the extra uncertainty. As n grows the t-distribution converges to normal, so the choice matters most for small samples (n < 30).
Why does increasing the confidence level widen the interval?
A higher confidence level requires a larger critical value (z* or t*), which directly multiplies the margin of error. You are asking for more certainty, so you pay with precision.
Why is there a warning about the normal approximation for proportions?
The z-interval for a proportion relies on the normal approximation to the binomial. This approximation requires np̂ ≥ 5 and n(1−p̂) ≥ 5. When either condition fails — especially for extreme proportions or very small n — the interval is unreliable. Consider an exact Clopper-Pearson interval instead.