Confidence Interval Calculator — Mean, Proportion & Margin of Error

When you hear "a poll found 52% support, ±3%" — that "±3%" is a confidence interval. It's an honest acknowledgment that sample data gives you an estimate, not a certainty, and here's how wide the uncertainty is. The CalcHub Confidence Interval Calculator computes intervals for both means and proportions, for any confidence level you choose.

Two Types of Confidence Intervals

1. Interval for a population mean — when your data is continuous measurements (heights, weights, response times). 2. Interval for a proportion — when your data is a percentage or count (approval ratings, defect rates, survey responses).

For a Population Mean

Formula:

CI = x̄ ± z* × (σ / √n) [known population SD]

CI = x̄ ± t* × (s / √n) [estimated from sample SD]

Where:

• x̄ = sample mean


• σ or s = standard deviation


• n = sample size


• z or t = critical value for your confidence level

Example: 50 patients, mean recovery time = 14.2 days, SD = 3.1 days, 95% CI:

z* = 1.96 (for 95%)

Margin of error = 1.96 × (3.1 / √50) = 1.96 × 0.438 = 0.86

CI: 13.34 to 15.06 days

For a Proportion

Formula:

CI = p̂ ± z* × √(p̂(1 − p̂) / n)

Example: 400 voters surveyed, 52% favor a candidate, 95% CI:

p̂ = 0.52, n = 400

SE = √(0.52 × 0.48 / 400) = √(0.000624) ≈ 0.02498

Margin of error = 1.96 × 0.025 = 0.049

CI: 47.1% to 57.1% (roughly ±5%)

Critical Values by Confidence Level

Confidence Level	z* Value	What It Means
90%	1.645	You'd be wrong 1 in 10 times
95%	1.960	You'd be wrong 1 in 20 times
99%	2.576	You'd be wrong 1 in 100 times

Higher confidence → wider interval. You can't get tighter bounds AND higher certainty without increasing sample size.

How to Use the Calculator

1. Select Mean or Proportion.
2. Enter: sample mean (or proportion), standard deviation (or leave blank for proportion), sample size.
3. Choose confidence level (90%, 95%, 99%, or custom).
4. Click Calculate — the interval, margin of error, and critical value all appear.

What a 95% Confidence Interval Actually Means

This is the most misunderstood part. A 95% CI does NOT mean "there's a 95% chance the true value falls in this interval." The true value either is in the interval or it isn't — probability doesn't apply to a fixed (if unknown) parameter.

What it actually means: if you repeated your study many times and calculated a 95% CI each time, about 95% of those intervals would contain the true value.

In practice, people treat it as "we're 95% confident the true value is somewhere in here," which is a workable interpretation even if statistically imprecise.

Effect of Sample Size

Sample Size	Margin of Error (95%, p=0.5)
50	±13.9%
100	±9.8%
400	±4.9%
1,000	±3.1%
2,500	±2.0%

Notice the diminishing returns — going from 100 to 400 quadruples the sample size but only halves the margin of error. This is because the formula uses √n, not n.

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What's the difference between a confidence interval and a prediction interval?

A confidence interval estimates where the population mean falls. A prediction interval estimates where a single new observation will fall. Prediction intervals are always wider because they account for individual variation, not just uncertainty about the mean.

When should I use t instead of z?

Use t* when sample size is small (typically n < 30) and/or you're estimating the population SD from your sample. The t-distribution has heavier tails than the normal, which gives you a wider (more conservative) interval to compensate for the extra uncertainty.

My 95% CI includes zero — what does that mean?

If you're calculating a CI for a difference or an effect, including zero means the data are consistent with "no effect." This is essentially the same conclusion you'd reach with a hypothesis test at the 5% significance level — the result is "not statistically significant."

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Related calculators: Z-Score Calculator · Sample Size Calculator · Standard Deviation Calculator