Normal Distribution Calculator

The normal (Gaussian) distribution is the bell curve: symmetric around its mean μ, with spread controlled by its standard deviation σ. Its density is f(x) = (1 / σ√(2π)) · exp(−(x−μ)² / 2σ²).

The central limit theorem says sums and averages of many independent random variables tend toward a normal distribution regardless of the underlying shape. That's why the normal models so many real-world phenomena — measurement errors, sample means, sums of small effects.

Any normal can be converted to the standard normal (μ=0, σ=1) via z = (x − μ)/σ. Probabilities and quantiles look up the same table for every problem.

P(X < x) — left tail. The cumulative distribution function Φ((x−μ)/σ). For x = μ this is always 0.5.

P(X > x) — right tail. 1 − Φ((x−μ)/σ).

P(a < X < b) — two-sided band. Φ((b−μ)/σ) − Φ((a−μ)/σ). The 68-95-99.7 rule comes from a = μ−kσ, b = μ+kσ for k = 1, 2, 3.

Given a target left-tail probability p, find the x such that P(X ≤ x) = p. Formally x = μ + σ · Φ⁻¹(p).

Used to set control limits (manufacturing), grade cutoffs (education), and stop-loss thresholds (finance). Example: the 99th percentile of daily portfolio losses is the value-at-risk (VaR) figure.

Common critical values: Φ⁻¹(0.95) ≈ 1.645, Φ⁻¹(0.975) ≈ 1.960, Φ⁻¹(0.99) ≈ 2.326, Φ⁻¹(0.995) ≈ 2.576.

Daily orders at a coffee shop are roughly normal with μ = 320 and σ = 40.

P(X > 400) = 1 − Φ((400−320)/40) = 1 − Φ(2.0) ≈ 1 − 0.9772 = 0.0228 → about 2.3% of days top 400 orders.

What's the 95th-percentile daily volume the staff should plan for? x = 320 + 40 · 1.645 ≈ 386 orders.

Skewed distributions (income, file sizes) — normal underestimates the right tail. Try a log-normal or report median + IQR instead.

Bounded variables (proportions in [0,1], counts ≥ 0) can be approximately normal only when far from the bounds.

Heavy-tailed phenomena (stock returns, network packet sizes) have more extreme events than a normal predicts. Student-t or stable distributions are common alternatives.

In education, IQ scores are defined as z-scores scaled to μ = 100 and σ = 15. An IQ of 130 is z = 2.0 — roughly the 98th percentile. SAT and GRE scores use similar scaling, allowing comparison across different test versions.

In medicine, pediatric growth charts plot height and weight as z-scores against age-matched norms. A child at z = −2.0 for height is below the 2.3rd percentile and may trigger referral for endocrine evaluation.

In manufacturing, Six Sigma defines process capability as the number of standard deviations between the process mean and the nearest specification limit. A '3-sigma' process has 2,700 defects per million; '6-sigma' aims for 3.4 defects per million by shrinking variation.