Normal Distribution Calculator

Model probability below a value, above a value, between two values, or the cutoff for a target percentile under a normal-distribution assumption.

How It Works

Formula

z = \frac{x - \mu}{\sigma}

\Phi(z) = 0.5 \cdot (1 + \operatorname{erf}(z / \sqrt{2}))

P(X \le x) = \Phi\!\left(\frac{x - \mu}{\sigma}\right)

P(X \ge x) = 1 - \Phi\!\left(\frac{x - \mu}{\sigma}\right)

P(a \le X \le b) = \Phi\!\left(\frac{b - \mu}{\sigma}\right) - \Phi\!\left(\frac{a - \mu}{\sigma}\right)

x = \mu + z\sigma, \quad z = \Phi^{-1}(p)

Variables, symbols and units

$x$: Raw measurement value or cutoff in the original units
$\mu$: Mean of the modeled normal distribution
$\sigma$: Standard deviation of the modeled normal distribution
$z$: Standardized distance from the mean, in standard-deviation units
$\Phi(z)$: Standard normal cumulative distribution function
$p$: Target cumulative percentile, written as a decimal probability

Calculation method explained

Choose the question shape first, then enter the mean and standard deviation in the same raw units as the measurement. Below x asks for the left-tail probability, above x asks for the right-tail probability, between a and b asks for the probability inside an interval, and percentile to cutoff asks which raw value corresponds to a cumulative percentile.

Standardize raw values with $z = \frac{x - \mu}{\sigma}$ so the question can be read on the standard normal curve.
For below, above, and between modes, convert the standardized value or bounds with $\Phi(z) = 0.5 \cdot (1 + \operatorname{erf}(z / \sqrt{2}))$ .
For percentile to cutoff, start from the cumulative percentile $p$ , use a stable inverse-normal approximation to get $z = \Phi^{-1}(p)$ , then map back with $x = \mu + z\sigma$ .
This is a model helper, not a normality test and not a hypothesis-testing tool.

Frequently Asked Questions

When is a normal model appropriate?

Use this calculator when a bell-curve approximation is reasonable for the measurement you care about. It does not test whether your real data are normal; it only applies the normal model you specify.

What do below, above, between, and percentile mean here?

Below x gives the cumulative area to the left of a threshold. Above x gives the right tail beyond a threshold. Between a and b gives the probability mass inside an interval. Percentile to cutoff starts with a cumulative percentile and returns the raw measurement value that matches it.

Why does the calculator return z-scores?

A z-score shows how far a raw value sits from the mean in standard-deviation units. The calculator standardizes your raw value first, then uses the standard normal curve to get the requested probability or cutoff.

How is this different from statistics, binomial probability, and scientific?

Statistics summarizes observed data. Binomial probability models yes/no counts across repeated trials. Scientific handles general-purpose calculations. This tool is narrower: it models continuous measurements under a normal-distribution assumption.

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