Gaussian Distribution Calculator

Complete tool for PDF, CDF, quantile, and Z-scores. Real-time computation & visualization for the bell curve. Ideal for data science, quality control, and finance.

X ~ N(μ, σ²)   with mean (μ) and standard deviation (σ > 0)
Compute density & cumulative probability at X
Inverse CDF: find x such that P(X ≤ x) = p
? Common distributions: Standard Normal (Z) IQ (μ=100, σ=15) Exam scores Narrow spread Process control
Computing distribution ...
Gaussian Results (real‑time)
PDF f(x)
0.2420
density at X
CDF F(x)
0.8413
P(X ≤ x)
Z‑score
1.0000
(x - μ)/σ
Quantile (p)
1.0000
x for given p
Tail probability P(X > x): 0.1587
Empirical rule (±1σ, ±2σ, ±3σ): 68.3%, 95.4%, 99.7%
PDF curve X = 1.00 Quantile (p) Mean μ

Understanding the Normal Distribution

The Gaussian (normal) distribution is the most important probability distribution in statistics, characterized by its bell shape and defined by mean μ and standard deviation σ.

? PDF & CDF

Probability Density:
f(x) = 1/(σ√(2π)) · e-(x-μ)²/(2σ²)

Cumulative Distribution:
F(x) = Φ((x-μ)/σ), with Φ the CDF of standard normal.

? Quantile (Inverse CDF)

Given probability p, the quantile x = μ + σ·Φ⁻¹(p). Used for confidence intervals, VaR, and hypothesis tests.

Z-score: (x-μ)/σ — number of standard deviations away from mean.

68–95–99.7 Rule (Empirical Rule)

±1σ: 68.27% ±2σ: 95.45% ±3σ: 99.73%

For a normal distribution, nearly all values lie within three standard deviations of the mean. This calculator visualizes these intervals on demand.

Frequently Asked Questions

They are identical. "Gaussian" honors Carl Friedrich Gauss, while "normal" refers to its common occurrence in nature.

We use a high‑precision rational approximation (erfinv) with Newton iteration, accurate to 6 decimal places for most p values.

σ = 0 would degenerate the distribution to a single point (Dirac delta). For meaningful continuous probabilities, σ must be positive.

Z = (x - μ)/σ. It tells how many standard deviations an observation x is from the mean. Used to compare different normal distributions.
? Pro features: Real‑time graph updates with vertical lines for your X value, the quantile (from probability p), and the mean. Hover over points for exact coordinates.