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Normal Distribution Calculator

Computes cumulative probabilities P(X < x), P(X > x) and P(x < X < x2) and the density for a normal (Gaussian) distribution with a given mean and standard deviation.

When to use

You need the probability that a normally distributed quantity falls below, above or between given values, or the z-scores of those values.

Do not use when: The variable is a count of successes or events (use binomial-distribution or poisson-distribution), or you need the value at a given probability from real data (use percentile).

Formula

z = (x − mean) / std_dev; P(X < x) = Φ(z); P(X > x) = 1 − Φ(z); P(x < X < x2) = |Φ(z2) − Φ(z)|; f(x) = exp(−z² / 2) / (std_dev · √(2π))

Φ is computed from the Abramowitz & Stegun erf approximation (absolute error < 1.5·10⁻⁷); results are exact to the displayed decimals. For a continuous distribution P(X ≤ x) = P(X < x).

Inputs

Parameter	Type	Required	Description
`mean`	number	default 0	Mean of the distribution (default 0 = standard normal).
`std_dev`	number	default 1	Standard deviation of the distribution (default 1 = standard normal). Range: > 0
`x`	number	yes	The value at which to evaluate the distribution.
`x2`	number	no	Optional second value; when given, the probability between x and x2 is also returned.

Outputs

Output	Type	Description
`z_score`	number	(x − mean) / std_dev.
`probability_below`	number	Cumulative probability up to x.
`probability_above`	number	Upper-tail probability beyond x.
`probability_density`	number	Value of the probability density function at x (not a probability).
`z_score_2`	number	(x2 − mean) / std_dev (only when x2 is given).
`probability_between`	number	Probability between the two values, in either order (only when x2 is given).

Example

Standard normal, x = 1.96: {"x":1.96} → {"z_score":1.96,"probability_below":0.975,"probability_above":0.025,"probability_density":0.05844}

IQ 85 to 115 (mean 100, SD 15): {"mean":100,"std_dev":15,"x":85,"x2":115} → {"z_score":-1,"probability_below":0.1587,"probability_above":0.8413,"probability_density":0.01613,"z_score_2":1,"probability_between":0.6827}

GET https://tttkmbb.com/api/v1/calculate/normal-distribution?x=1.96

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/normal-distribution (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/normal-distribution · Markdown: https://tttkmbb.com/statistics/normal-distribution.md · JSON definition: https://tttkmbb.com/statistics/normal-distribution.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="normal-distribution"
OpenAPI operationId: calculate_normal_distribution_probability
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution (government)
Wikipedia – Normal distribution (reference)

FAQ

What is the 68–95–99.7 rule?

For a normal distribution about 68.27 % of values lie within 1 SD of the mean, 95.45 % within 2 SD and 99.73 % within 3 SD; the second example (85–115 with SD 15) reproduces the 68.27 % figure.

Why is the density larger than 1 sometimes?

f(x) is a density, not a probability: it can exceed 1 when the standard deviation is small (below 0.399). Probabilities come from areas under the curve, i.e. the CDF outputs.

Related calculators

Z-Score Calculator — Standardise a single value and read its percentile.
Confidence Interval Calculator — Use normal critical values to build an interval for a mean.