Normal Distribution PDF/CDF
Compute Normal(μ, σ) PDF and CDF at a point.
Normal PDF: the Gaussian density
The normal (Gaussian) probability density function is the formula behind the familiar bell-shaped curve: f(x) = (1 / (σ√(2π)))·e^(−(x − μ)² / (2σ²)). Its peak lands on the mean μ, and how wide it spreads depends on the standard deviation σ. One thing trips people up: the PDF returns density rather than probability. To get an actual probability you need the area under the curve (the CDF), which has no elementary closed form and gets computed through the error function erf. Standardizing with Z = (X − μ)/σ turns any normal into the standard normal N(0, 1). There's also the handy 68-95-99.7 rule of thumb: roughly 68% of the mass sits within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. As an example, at μ = 0, σ = 1, x = 1.96, f(x) ≈ 0.0584.
Applications
You see it in biometric measurements like height, weight and IQ, in measurement error across physics and engineering, and in finance (Black-Scholes assumes log-normal returns). It shows up in statistical process control, where Six Sigma targets 3.4 defects per million, and in regression analysis, since OLS assumes normal residuals. The Central Limit Theorem is the deeper reason behind all of this: add up enough independent variables and the result drifts toward a normal distribution.
FAQ
Why can f(x) exceed 1? Because density isn't the same thing as probability. A small σ makes the curve tall and narrow, so f(x) can climb above 1 with no problem. What's constrained to equal 1 is the integral, the total area.
PDF vs CDF — what is the difference? The PDF tells you the density at a single point, while the CDF gives the accumulated probability P(X ≤ x). Put another way, the CDF is just the integral of the PDF.
When is the normal a bad fit? Whenever the data is skewed, like income or time-to-failure, or when the tails are heavy, as financial returns tend to be during crises. In those cases a log-normal, Student's t, or an extreme value distribution usually fits better.
What does the Central Limit Theorem say? Take the average of many independent samples from any distribution with finite variance, and that average converges to a normal. That convergence is the reason the Gaussian keeps turning up in practice.
Related Tools
Z Table (Normal CDF)
Compute P(Z ≤ z) and P(|Z| ≤ z) for the standard normal distribution.
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Computes the prediction interval for a new individual observation, from a sample. Be careful not to confuse it with the confidence interval for the mean: that one estimates where the population mean lies; this one estimates where the next individual value you observe will fall — which is why it's always wider, carrying the uncertainty of the mean plus the natural variability of the data. The decisive factor is the √(1 + 1/n) under the t distribution. Enter the sample and the confidence level.
Kolmogorov-Smirnov Test (Two-Sample)
Computes the two-sample Kolmogorov-Smirnov test, which checks whether two samples come from the same distribution — without assuming what that distribution is. It compares the empirical cumulative distribution functions of the two samples and takes the largest vertical distance between them, the D statistic. Unlike the t-test, which only compares means, the KS test is sensitive to any difference in shape, spread or location. Enter the two samples of values.
Normal CDF Φ(z) Approximation
Enter a z-score to get Φ(z), the cumulative probability of the standard normal N(0,1), via the Abramowitz-Stegun error function approximation.
Normal Distribution Z and Percentile Calculator
Converts a normal distribution value to standardized z score and estimates the cumulative percentile using error function approximation.
Normal Distribution PDF Calculator
Calculate the probability density of the normal (Gaussian) distribution at a point, from the mean and standard deviation. Key in statistics and data science.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.