Normal Distribution Calculator
Enter mean (μ) and standard deviation (σ) of your normal distribution and compute PDF, CDF, tail probability, the probability between two values, and the inverse CDF (quantile). The shaded bell curve updates to show the selected area.
σ > 0
returns x such that P(X ≤ x) = p
Illustrative only: area under the curve refers to current calculation.
z-score
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PDF f(x)
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CDF P(X ≤ x)
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Area / Result
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Formulas used
PDF:
\( f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( - \frac{(x-\mu)^2}{2 \sigma^2} \right) \)
CDF:
\( F(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma \sqrt{2}} \right) \right] \)
Standardize: \( z = \frac{x-\mu}{\sigma} \)
Typical tasks
- Find P(X ≤ x) for a normal variable.
- Find P(a < X < b) by subtracting two CDF values.
- Find a critical value xα given a probability (inverse CDF).
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
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Formula (extracted text)
PDF: \( f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( - \frac{(x-\mu)^2}{2 \sigma^2} \right) \) CDF: \( F(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma \sqrt{2}} \right) \right] \) Standardize: \( z = \frac{x-\mu}{\sigma} \)
Variables and units
- No variables provided in audit spec.
Sources (authoritative):
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.