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Calculator Answers

Confidence Interval Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A confidence interval calculator is a statistical utility that computes the range of values within which a population parameter is estimated to lie. Using sample data (such as sample mean, sample size, and standard deviation) or sample proportions, the calculator constructs the lower and upper bounds of the interval at a specified confidence level (commonly ninety-five or ninety-nine percent). It supports z-distributions for large samples and t-distributions for smaller sample sizes. Researchers use this tool to quantify the uncertainty of their statistical estimates.

Pick mean or proportion mode and a confidence level. Enter the sample estimate, spread, and sample size, and the calculator returns the lower and upper bounds, the margin of error, and the standard error.

See also: P Value Calculator, Margin of Error Calculator, and Standard Deviation Calculator – Variance & SD.

Quick Answer

Compute the confidence interval for your sample data. Enter the sample mean, standard deviation, sample size, and confidence level to find the statistical range.

Confidence level

e.g. 100

Sample standard deviation. · e.g. 15

e.g. 50

This calculator uses the standard normal z-score approximation. For small samples or unknown population variance, statistical software typically applies a t-distribution. Treat this as a planning estimate.

Confidence interval (mean)

95% CI around 100

95.8422 to 104.1578

Margin of error: ±4.1578

Z-score used1.96
Standard error2.1213
Margin of error4.1578
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Examples

Mean: x̄ = 100, s = 15, n = 50, 95% confidence

≈ 95.84 to 104.16

Proportion: p = 60%, n = 500, 95% confidence

≈ 55.71% to 64.29%

Proportion: p = 50%, n = 1000, 99% confidence

≈ 45.93% to 54.07%

How it works

A confidence interval is the sample estimate plus or minus a margin of error. Margin of error is the z-score for the chosen confidence level times the standard error. Standard error depends on whether the estimate is a mean or a proportion.

Mean SE · s ÷ √n

Proportion SE · √(p(1 − p) ÷ n)

CI · estimate ± z × SE

Z-scores: 90% = 1.645 · 95% = 1.96 · 99% = 2.576

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P Value CalculatorConvert a z-score to a p-value for a one-tailed or two-tailed test, with the area on each side of the curve.Margin of Error CalculatorCalculate margin of error from sample size, confidence level, and sample proportion. Returns the standard error, z-score, and the confidence range.Standard Deviation Calculator – Variance & SDUse this sd calculator to compute sample and population standard deviation, variance, mean, and count from a list of numbers.

Frequently asked questions

A confidence interval is a range around a sample estimate that should contain the true population value most of the time, under repeated sampling. A 95 percent confidence interval, in plain terms, says that if you repeated the same sampling process many times, about 95 percent of the intervals built this way would capture the real population value. The specific interval you got either contains the truth or does not; the 95 percent describes the procedure, not the single result.

Compute the standard error: for a mean it is the sample standard deviation divided by the square root of n; for a proportion it is the square root of p(1 − p) divided by n. Multiply by the z-score for the confidence level (1.645 at 90 percent, 1.96 at 95 percent, 2.576 at 99 percent) to get the margin of error. Add and subtract it from the sample estimate to get the interval.

Use mean mode when the data is continuous (heights, scores, dollar amounts) and you have a sample mean and standard deviation. Use proportion mode when the data is binary (yes/no, success/failure) and you have a sample proportion. The two modes use different standard-error formulas because the underlying distributions are different.

The z-score approximation is appropriate when the population standard deviation is known or when sample sizes are large (say, above 30 for means). For small samples or unknown population variance, real statistical software typically uses a t-distribution, which produces slightly wider intervals. This calculator uses the z approximation, which is the convention for most planning-stage and intro-stats use.

Use this calculator as a planning estimate or for educational use. Formal research analysis needs full attention to sample design (random, stratified, clustered), non-response, weighting, and the right test for the data type. For published results, use professional statistical software and confirm assumptions with a qualified statistician.

Wider for higher confidence (99 percent intervals are wider than 95 percent), wider for smaller samples, and wider for more variable data. Doubling the sample size cuts the margin of error by about 30 percent. The same data at 99 percent confidence has a margin about 31 percent larger than the same data at 95 percent confidence (2.576 vs 1.96).