The Gaussian Curve

The normal distribution describes many natural phenomena, from measurement errors to heights of people. Its bell-shaped curve is defined by the density function

$f\left(x\right)=\frac{1}{\mathrm{\backslash sqrt\{2\backslash pi\}}\mathrm{\backslash sigma}}{e}^{-\frac{{x-\mathrm{\backslash mu}}^2}{2{\mathrm{\backslash sigma}}^2}}$

The cumulative distribution function (CDF) gives the probability that a random variable is less than $x$. It is expressed in terms of the error function $\mathrm{\backslash operatorname\{erf\}}$, which integrates the bell curve.

Why Use This Calculator?

Statistics, physics, finance, and many other fields rely on the normal distribution. When dealing with standard scores, confidence intervals, or noise models, you often need quick access to its PDF and CDF values. This tool computes them for any mean $\mathrm{\backslash mu}$ and standard deviation $\mathrm{\backslash sigma}$.

Example

If $\mathrm{\backslash mu}=0$, $\mathrm{\backslash sigma}=1$, and $x=1$, the PDF evaluates to $0.24197$ and the CDF to $0.84134$. You can verify these values in standard statistical tables.

How It Works

The script below uses math.js for accurate evaluation of the error function. By substituting your values into the formulas, it returns the density and cumulative probability rounded to six decimals.