What is a confidence interval?

A confidence interval is a range built from sample data that is likely to contain the true population mean. The confidence level describes the method, not one interval: if you repeated the study many times, about 95% of the 95% intervals would contain the true mean. Any single interval either contains it or does not.

When is a Z-score used instead of a T-score?

Strictly, z is for a known population standard deviation and t is for a standard deviation estimated from the sample, at any sample size. This calculator follows the common textbook shortcut: t with n − 1 degrees of freedom below n = 30 and z from n = 30 up. At n = 30 the exact t value is 2.045 instead of 1.960, so the interval shown is about 4% narrower than the exact t interval. The gap shrinks as n grows (t = 2.010 at n = 50).

What does the margin of error represent?

The margin of error is the half-width of the confidence interval. It tells you how far the sample mean could be from the true population mean while still being consistent with your data at the chosen confidence level.

Does a higher confidence level always mean a better interval?

Not necessarily. A higher confidence level produces a wider interval, which is less precise. A 99% CI is more likely to contain the true value but gives a broader, less actionable range than a 95% CI. The right choice depends on the trade-off between certainty and precision you need.

What if I only have population standard deviation?

Enter σ in the Standard Deviation field together with your real sample size. Do not change n, because the standard error σ ÷ √n depends on it. For n of 30 or more the calculator already uses z, the correct method when σ is known. For n below 30 it uses t, which gives a slightly wider interval than the z interval x̄ ± 1.960 × σ ÷ √n at 95%.

What z value is used for a 90%, 95% or 99% confidence interval?

The two-sided z values are 1.645 for 90%, 1.960 for 95% and 2.576 for 99%. The calculator uses them from n = 30 up. Below 30 it uses t instead, for example t = 2.262 for a 95% interval with n = 10 (9 degrees of freedom).

Can it calculate a confidence interval for a proportion?

No, it handles a mean only. For a proportion, use p̂ ± z × √(p̂(1 − p̂) ÷ n). With 520 yes answers out of 1,000, p̂ = 0.52, the standard error is 0.0158 and the 95% interval is 0.52 ± 1.960 × 0.0158, or 48.9% to 55.1%.

Confidence Interval Calculator

Enter your sample data and confidence level to instantly compute the confidence interval and margin of error.

CI = x̄ ± critical value × s ÷ √n.

Sample Parameters

Sample Size (n)

Sample Mean (x̄)

Standard Deviation (s or σ)

Confidence Level

Enter your sample parameters on the left and click Calculate.

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Quick answer

CI = x̄ ± critical value × s ÷ √n. This calculator uses a t value with n − 1 degrees of freedom when n is below 30, and the z value (1.645, 1.960 or 2.576 for 90%, 95% or 99%) when n is 30 or more. For n = 10, mean 50 and standard deviation 8, the 95% interval is 50 ± 2.262 × 2.5298, or 44.28 to 55.72.

Examples

n, mean, SD	95% value	95% CI
10, 50, 8	t = 2.262	44.28–55.72
25, 100, 15	t = 2.064	93.81–106.19
30, 100, 15	z = 1.960	94.63–105.37
50, 75.4, 10.2	z = 1.960	72.57–78.23
100, 50, 10	z = 1.960	48.04–51.96
400, 50, 10	z = 1.960	49.02–50.98

Enter n, mean and standard deviation, then click Calculate.

Summary

A confidence interval estimates the range within which a population parameter likely falls, based on sample data. This calculator supports the most common confidence levels (90%, 95%, 99%) and uses the Z-distribution for large samples or the T-distribution for small samples (n < 30). Enter your sample statistics and get the lower bound, upper bound, and margin of error in one click.

How it works

Enter the sample size (n), sample mean, and standard deviation.
Select your desired confidence level (90%, 95%, or 99%).
The calculator chooses Z or T critical value based on sample size.
Margin of error is computed as critical value times standard error.
The confidence interval is displayed as mean plus or minus the margin of error.

Use cases

Estimate population mean from survey or experiment data.
Validate statistical hypotheses in academic research.
Determine acceptable ranges for quality control measurements.
Calculate margin of error for poll results.
Assess precision of lab measurements and sensor readings.
Support data-driven decision making in business analytics.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu