Calculators
Normal Distribution Calculator
Calculate normal (Gaussian) distribution density, cumulative and upper-tail probabilities, quantiles, and standard scores privately in your browser.
Model continuous data with the bell curve.
The normal distribution models continuous quantities that cluster around a mean μ with standard deviation σ > 0. The density is f(x) = 1/(σ√2π) × e^(-(x-μ)²/(2σ²)). Cumulative probability is P(X ≤ x) = Φ((x-μ)/σ) and upper-tail is P(X > x) = 1 - Φ((x-μ)/σ), where Φ is the standard normal CDF. The quantile x for probability p is x = μ + σ × Φ⁻¹(p). For μ = 0, σ = 1 and x = 1, the density is about 0.24197, cumulative probability is about 0.84134, standard score z = 1, and upper-tail is about 0.15866. Enter mean μ and standard deviation σ > 0 from -1e12 to 1e12, x from -1e12 to 1e12, and probability p from 0 to 1 (exclusive); scientific notation is accepted. Results use floating-point arithmetic rounded to 12 significant digits; extreme tails may round to zero or one. This is a model, not a fit to observed data. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the difference between the probability and quantile modes?
In probability mode, you enter a value x and get the cumulative probability P(X ≤ x), upper-tail probability P(X > x), density f(x), and standard score z. In quantile mode, you enter a cumulative probability p (between 0 and 1) and get the value x such that P(X ≤ x) = p, along with the corresponding standard score z.
What is the standard score (z-score)?
The standard score is z = (x − μ) / σ. It measures how many standard deviations x is from the mean. For the standard normal distribution (μ = 0, σ = 1), the standard score equals x. The calculator always reports z so you can relate results to standard normal tables.
Is the density a probability?
No. Density has inverse measurement units and can exceed one, for example with a small σ near the mean. A single exact value has probability zero in this continuous model; probabilities come from areas under the density, such as P(X ≤ x).
How does this relate to confidence intervals?
A two-sided confidence interval for the mean uses the quantiles of the normal distribution. For 95% confidence, the quantiles are Φ⁻¹(0.025) ≈ −1.96 and Φ⁻¹(0.975) ≈ 1.96. The Confidence Interval Calculator uses this distribution for its normal-approximation method.
When is the normal model appropriate?
Use it for quantities that cluster symmetrically around a central value with a known or estimated standard deviation, such as measurement errors, heights, or test scores. It is not suitable for bounded, skewed, or multi-modal data, or for waiting times (use the exponential distribution) or counts (use the Poisson distribution).