Normal Distribution Calculator
Explore a specified normal model using its mean and standard deviation. Choose an area probability, percentile, or density calculation. Normal model only; supplied mean and deviation are assumptions, not verified exam norms.
A QUICK WALKTHROUGH
How to use this tool
- Enter a finite mean and positive standard deviation within the stated limits.
- Choose left tail, right tail, interval, percentile, or density; enter the relevant values.
- Calculate and read the approximate result together with its accuracy limitations.
- Cohort: one score or score | positive integer frequency per line
- Tie convention; I confirm this is the complete comparison cohort; Source / declared assumptions
- Named scenarios: name | query score per line; Copy report; Download TXT
Model and supported range
For X ~ N(μ, σ²), z = (x−μ)/σ. Mean must be finite; σ must be finite and positive; each value or interval bound must satisfy |z| ≤ 8, with a ≤ b. Percentiles are limited to 0.1%–99.9%. Density uses exp(−z²/2)/(σ√(2π)). No data is fitted or tested for normality.
Numerical approximation
CDF uses the Abramowitz–Stegun 7.1.26 error-function polynomial. Conservative absolute probability error before display rounding is ≤ 1.5×10⁻⁷ per tail and ≤ 3×10⁻⁷ for an interval. Very small probabilities may have large relative error. Percentiles are found by 50 bisection steps on this approximate CDF in [−8,8]; this does not remove the CDF error. Value uncertainty is amplified in tails and scales with σ. Displayed digits are not guaranteed accurate digits.
Empirical cohort
Empirical percentages use the complete supplied cohort. Strict: below/N; inclusive: (below+ties)/N; midrank: (below+ties/2)/N. No external norm data.
Normal Distribution Calculator
Normal numeric input: decimal notation with optional exponent, at most 100 characters. Values must be finite; nonzero inputs that underflow to zero are rejected. Report numbers retain the full JavaScript round-trip value; the normal approximation error still applies.
GOOD TO KNOW
Common questions
Is probability density a probability?
No. Density is curve height and can exceed one. Probabilities are areas: left tail P(X ≤ x), right tail P(X ≥ x), and interval P(a ≤ X ≤ b). For a continuous normal model, including or excluding an endpoint gives the same probability.
Are values sent to a server?
No. Calculations run in your browser. Results describe the chosen normal model, not verified properties of an observed dataset.
Tie convention?
Empirical percentages use the complete supplied cohort. Strict: below/N; inclusive: (below+ties)/N; midrank: (below+ties/2)/N. No external norm data. Normal model only; supplied mean and deviation are assumptions, not verified exam norms.