What is a confidence interval?

A range of values computed from sample data that is likely to contain the true population parameter at a given confidence level.

Why is 95% the most common confidence level?

Convention. It balances precision (narrow interval) with confidence (high certainty). 99% is wider; 90% is narrower.

How do I reduce the margin of error?

Increase sample size n (MOE ∝ 1/√n) or reduce the confidence level. Doubling n roughly halves the MOE.

Confidence Interval Calculator

Enter your sample statistics to compute a confidence interval: for a mean (using the Z distribution) or for a proportion. Margin of error and standard error are shown for each result.

How to use this calculator

Select whether you are estimating a mean or a proportion.
Choose the confidence level (95% is standard for most research).
Enter your sample statistics: mean and SD (or proportion) and sample size.
Read the confidence interval bounds and margin of error.

Formula & method

Z-based confidence interval formula

CI = x̄ ± z* × (s/√n) · For proportions: CI = p̂ ± z* × √(p̂(1−p̂)/n)

z* critical values: 90%=1.645, 95%=1.96, 99%=2.576. SE = s/√n for means.

Example

x̄=100, s=15, n=50, 95% CI: SE=15/√50≈2.12, MOE=1.96×2.12≈4.16. CI=[95.84, 104.16].

Key insights

A 95% CI means: if you repeated the study 100 times, 95 of the CIs would contain the true parameter.
Wider CIs = more uncertainty. Narrower CIs require larger samples.
Margin of error (MOE) = z* × SE. Higher confidence levels require larger z* values.
For proportions, SE = √(p(1−p)/n). Maximum SE occurs at p = 0.5 (most uncertainty).

How to interpret your result

Reading the interval

CI [A, B] means the true population parameter is estimated to lie between A and B with the stated confidence level.

Confidence vs probability

The true mean is fixed (not random). The CI is random (it depends on your sample). The confidence level refers to the procedure, not the specific interval.

Frequently asked questions

What is a confidence interval?: A range of values computed from sample data that is likely to contain the true population parameter at a given confidence level.
Why is 95% the most common confidence level?: Convention. It balances precision (narrow interval) with confidence (high certainty). 99% is wider; 90% is narrower.
How do I reduce the margin of error?: Increase sample size n (MOE ∝ 1/√n) or reduce the confidence level. Doubling n roughly halves the MOE.

Related calculators

Z-score Calculator — Convert a raw score to a Z-score (standard score), find its percentile, and compute cumulative probability under the normal distribution.
Sample Size Calculator — Determine the minimum sample size needed for a survey or study based on margin of error, confidence level, and population size.
P-value Calculator — Calculate the P-value from a Z-statistic for one-tailed or two-tailed hypothesis tests and determine statistical significance.

See all math calculators · Explore all online calculators