Confidence Interval Calculator

Calculate confidence intervals for a mean or proportion with margin of error and step-by-step explanations

Sample Mean (x̄)

Sample Std Dev (s)

Sample Size (n)

Confidence Level

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Common Confidence Levels Reference

Confidence Level	Z-Score	Interpretation
90%	1.645	10% chance of error
95%	1.960	5% chance of error
99%	2.576	1% chance of error
99.9%	3.291	0.1% chance of error

Copy-ready result handoff

Use this after estimating an interval from sample statistics.

Common searches

confidence interval calculatormargin of errorsample mean

Input check

Confirm sample size, sample mean or proportion, and confidence level.

Assumption

The interval is an estimate, not a guarantee that a future value will fall inside it.

Confidence interval note

Copy this after replacing the bracketed result with the value shown by the calculator.

Confidence interval result: [paste lower bound, upper bound, margin of error, and confidence level].
Inputs checked: Confirm sample size, sample mean or proportion, and confidence level.
Assumption: The interval is an estimate, not a guarantee that a future value will fall inside it.
Use case: Survey report, lab estimate, product sample review, or measurement uncertainty.
Next check: Use Sample Size Calculator when the margin of error is too wide.

Choose mean or proportion mode, enter the sample statistic, sample size, spread if needed, and confidence level. The page returns the lower bound, upper bound, margin of error, and point estimate side by side.

Formula check for confidence intervals

mean CI = x_bar +/- critical * s / sqrt(n); proportion CI = p_hat +/- z * sqrt(p_hat*(1-p_hat)/n)

Worked example: x_bar = 75, s = 10, n = 50, and 95% confidence gives SE = 10 / sqrt(50) = 1.414, margin = 1.96 * 1.414 = 2.77, so the interval is about 72.23 to 77.77.

Variables and units

x_bar or p_hat: sample mean or sample proportion, in the same scale as the data.
s: sample standard deviation for mean intervals, in the original data unit.
n: sample size; larger n narrows the standard error.
critical value: t or z multiplier selected from the confidence level.

Assumptions and limits

Sample observations should be independent and representative.
Mean intervals assume the sample standard deviation is a reasonable estimate of spread.
Proportion intervals need enough successes and failures for the normal approximation.

Common mistakes

Reading a 95% interval as a 95% probability statement about this one fixed parameter.
Using z when a small-sample mean interval should use the t critical value.
Treating margin of error as the full interval width instead of half the width.

Confidence Interval Inputs, Assumptions, And Limits

What this calculator estimates

A confidence interval combines a point estimate with a margin of error. For a mean, the calculator uses the sample mean, sample standard deviation, sample size, and a critical value. For a proportion, it uses the sample percentage, sample size, and a z critical value.

How to read the interval

The lower and upper bounds show a plausible range for the population parameter under the selected method. A wider interval means more uncertainty. Increasing the confidence level widens the interval, while increasing sample size usually narrows it.

Mean intervals vs proportion intervals

Use mean intervals for measured numeric data, such as time, height, or test scores. Use proportion intervals for percentages from yes/no outcomes, such as survey support or defect rate. If you need the sample size before collecting data, use the Sample Size Calculator.

Before relying on the answer

Confirm the sample was collected in a way that represents the population you care about. For mean intervals, make sure the spread value is in the same unit as the mean. For proportions, avoid using the normal approximation when successes or failures are very rare.

References & Sources

Questions about this tool

What does a 95% confidence interval mean?: It means the method would capture the true parameter in about 95% of repeated samples. A single completed interval either contains the true value or it does not.
Should I use the mean mode or the proportion mode?: Use mean mode for numeric measurements such as scores, weights, or times. Use proportion mode for yes/no or success/failure outcomes reported as a percentage.
Is margin of error the full interval width?: No. Margin of error is the half-width. Add and subtract it from the point estimate to get the lower and upper bounds.
Why does a larger sample size narrow the interval?: Larger samples reduce standard error, so the same confidence level can be reached with a smaller margin of error.

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