Z-Score & Confidence Interval Calculator

This calculator covers two related jobs from introductory statistics. Z-score mode standardizes one raw value x against a normal distribution described by its mean μ and standard deviation σ: z = (x − μ) / σ. The result tells you how many standard deviations x sits above or below the mean, and the tool also reports the cumulative probability Φ(z) and the one- and two-tailed tail areas beyond it — every number substituted into the formula, not just handed to you as a final answer.

The probability comes from a genuine closed-form approximation of the standard normal cumulative distribution function, the Abramowitz & Stegun 7.1.26 rational approximation of the error function (accurate to about 1.5 × 10⁻⁷), not a rounded lookup table. For x = 82, μ = 70, σ = 8, the z-score is exactly 1.5, and the approximation gives Φ(1.5) ≈ 0.9332 — the textbook value — with a one-tailed tail probability of about 0.0668 and a two-tailed probability of about 0.1336.

Confidence interval mode answers a different question: given a sample mean x̄, a standard deviation σ and a sample size n, what range plausibly contains the true population mean? The calculator looks up the exact critical value z* for the confidence level you choose (80% → 1.282, 90% → 1.645, 95% → 1.96, 98% → 2.326, 99% → 2.576) from a bundled standard-normal table, computes the standard error SE = σ / √n, the margin of error E = z* × SE, and the interval x̄ ± E.

A worked example: for a sample mean of 100, a standard deviation of 15 and a sample size of 36 at 95% confidence, SE = 15 / √36 = 2.5, the margin of error is 1.96 × 2.5 = 4.9, and the confidence interval is [95.1, 104.9] — every one of those numbers shown on the page, not compressed into a single answer. Both modes run entirely client-side, so the values you enter never leave your browser.