# 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.

- Calculator id: `normal-distribution` · Category: Statistics & Probability (`statistics`) · Tool name: `calculate_normal_distribution_probability`
- Canonical page: https://tttkmbb.com/statistics/normal-distribution · This document: https://tttkmbb.com/statistics/normal-distribution.md · JSON definition: https://tttkmbb.com/statistics/normal-distribution.json

## Purpose

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.

**Use when:** 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).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `mean` | number |  | optional, default 0 | Mean of the distribution (default 0 = standard normal). |
| `std_dev` | number |  | optional, default 1 | Standard deviation of the distribution (default 1 = standard normal). (> 0) |
| `x` | number |  | required | The value at which to evaluate the distribution. |
| `x2` | number |  | optional | Optional second value; when given, the probability between x and x2 is also returned. |

## Output

| Field | Type | Unit | 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). |

## 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).

## Data Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution — https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (government, retrieved 2026-09-23)
- Wikipedia – Normal distribution — https://en.wikipedia.org/wiki/Normal_distribution (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/normal-distribution?x=…`
- `POST https://tttkmbb.com/api/v1/calculate/normal-distribution` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/normal-distribution · OpenAPI operationId `calculate_normal_distribution_probability` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "normal-distribution", "inputs": {…}}`

## Example

- Standard normal, x = 1.96: inputs `{"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): inputs `{"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
```

## Limitations

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). 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). All values are computed from the formula above; no measurement or live data is involved.

## 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

- [Z-Score Calculator](https://tttkmbb.com/statistics/z-score.md) — Standardise a single value and read its percentile.
- [Confidence Interval Calculator](https://tttkmbb.com/statistics/confidence-interval.md) — Use normal critical values to build an interval for a mean.