ZeroServer.tools

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.

Private & free — this tool runs entirely in your browser.