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  4. Confidence Interval Calculator

Confidence Interval Calculator

z confidence interval for a mean (x̄ ± z·σ/√n) or Wald interval for a proportion, with SE, margin, and multiple interval forms.

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How it works

Half-width is z times the standard error. Higher confidence widens the interval; larger n narrows it.

x̄ ± z·(σ/√n) · p̂ ± z√(p̂(1−p̂)/n)

When to use

Use this tool when you need to: calc.net demo: n=100, x̄=22.8, σ=2.7, 95% → about 22.8 ± 0.529. Or n=100, mean=50, sd=10 → [48.04, 51.96].

How to use this tool

  1. Enter your values in the fields above.
  2. Review the formula and any mode options for your problem.
  3. Read the result and the step-by-step solution.

Calc.net demo: n=100, x̄=22.8, σ=2.7, 95% → about 22.8 ± 0.529. Or n=100, mean=50, sd=10 → [48.04, 51.96].

Frequently asked questions

What is the Confidence Interval Calculator?
Half-width is z times the standard error. Higher confidence widens the interval; larger n narrows it.
How do I use the Confidence Interval Calculator?
Enter the required values, then calculate. The Confidence Interval Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Confidence Interval Calculator use?
x̄ ± z·(σ/√n) · p̂ ± z√(p̂(1−p̂)/n)

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Confidence interval calculator

Estimate a normal-based confidence interval (or margin of error) for a sample mean with known σ — or a Wald interval for a proportion. Assumes the sample mean is approximately normal.

Interval for

Confidence level

Uses x̄ ± z·(σ/√n). Population σ known, or n large enough that sample s ≈ σ. For raw data only, compute mean and SD first. Confidence as 95 or 0.95.

Z-values for common confidence levels
Confidence levelz value
70%1.036
75%1.15
80%1.282
85%1.44
90%1.645
95%1.96
98%2.326
99%2.576
99.5%2.807
99.9%3.291
99.99%3.891
99.999%4.417

Reference table (calculator.net style). The calculator computes z from the inverse normal of your exact confidence level.

Try an example

How it works

What is a confidence interval?

A confidence interval is a range of estimates for an unknown parameter (often a population mean), built from sample data at a pre-chosen confidence level such as 95%. That level describes the long-run success rate of the procedure — if you repeated the study many times, about 95% of such intervals would contain the true value. It is not a probability that any single realized interval contains the parameter; once computed, the interval either covers μ or it does not.

Mean intervals (known σ)

When the sampling distribution of x̄ is normal with known σ (or n is large so s ≃ σ), use x̄ ± z·(σ/√n). Width grows with higher confidence and shrinks with larger n or smaller σ. This tool does not use the t-interval for unknown σ with small n — use a t-based procedure in that case.

How intervals are written

Equivalent forms include center ± margin (22.8 ± 0.529), relative percent of the center (22.8 ± 2.3%), or an explicit range [22.271 – 23.329].

Proportion intervals

For a sample proportion, the Wald interval is p̂ ± z√(p̂(1−p̂)/n). Prefer large n; near 0/1, consider more advanced intervals.

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Probability, counting, sampling design, and normal-based inference tools work together. Descriptive statistics calculators help when you start from raw data.

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