Confidence Interval Calculator

Estimate confidence intervals for sample statistics or raw numeric lists. Features visual distribution charts.

Inputs

Interval type

Confidence Level (%)

Sample size (n)

Sample mean (x̄)

Sample std dev (s)

Step-by-Step Substitution

1. Calculate Significance Level (α): α = 1 - 95% = 0.0500

2. Half-alpha (α/2) = 0.0250

3. Degrees of freedom (df): df = n - 1 = 29

4. Find critical t-value: t*(df, 1 - α/2) = t*(29, 1 - 0.0250) = 2.0456

5. Calculate Standard Error (SE): SE = s / √n = 10 / √30 = 1.8257

6. Margin of Error (ME): ME = t* * SE = 2.0456 * 1.8257 = 3.7347

7. Interval Bounds: [Mean - ME, Mean + ME] = [50 - 3.7347, 50 + 3.7347] = [46.2653, 53.7347]

95% Confidence Interval

Lower bound

46.2653

Upper bound

53.7347

Standard error:1.82574

Critical value:2.0456

Margin of error (ME):3.7347

How the confidence interval is computed

A confidence interval bounds a population parameter based on sample stats and confidence thresholds. When standard deviation $\sigma$ is known, we use the Z-distribution:SE = \sigma / \sqrt{n}. When $\sigma$ is unknown, we use the t-distribution with $df = n - 1$.

The proportion calculator utilizes Wald standard error: SE = \sqrt{p(1 - p) / n}. The margin of error equals Critical Value $\times$ Standard Error, setting the bounds around our sample mean.

Built and maintained by Meet Shah · Last updated 1 August 2026

What this tool is used for

Putting an interval around a survey result before quoting it as a number.
Checking whether two measured proportions actually differ or merely appear to.
Working out how large a sample needs to be for a useful interval.
Reporting a mean with its uncertainty rather than as a point estimate.
Sanity-checking a result that looks surprisingly precise for its sample size.

Frequently Asked Questions

What does a 95% confidence interval mean?: That if you repeated the sampling procedure many times, 95% of the intervals produced would contain the true parameter. It is a statement about the METHOD, not about this interval — the common reading, that there is a 95% chance the value is inside, is the Bayesian claim and not what this computes.
How does sample size affect the width?: The margin of error shrinks with the square root of n, so quadrupling the sample halves the interval. That relationship is why polls settle around a thousand respondents — the next meaningful narrowing costs four times as much.
When should I use t rather than z?: When the population standard deviation is unknown and estimated from the sample, which is nearly always. The t distribution has heavier tails to account for that extra uncertainty, and it converges on z as n grows — the difference is negligible past about 30.
What does it mean if two intervals overlap?: Less than people assume. Overlapping intervals do not imply the difference is insignificant — the correct test is on the difference itself, which has its own interval. Comparing two intervals by eye is systematically conservative and misses real effects.
Does a wide interval mean the study is wrong?: It means the estimate is imprecise, which is different. A wide interval honestly reports what a small or noisy sample can support, and it is far more informative than a point estimate reported alone — which is why journals increasingly require intervals rather than p-values.

Common errors and gotchas

Reading the interval as a 95% chance the true value is inside it, which is a different statement.
Using a proportion formula on a very small sample or an extreme proportion, where the approximation fails.
Assuming a random sample when the data came from a self-selected source.
Comparing two overlapping intervals and concluding no difference, which does not follow.
Reporting an interval while ignoring far larger non-sampling error in how the data was collected.