Normal Distribution Calculator
Solve common normal-distribution probability and percentile problems from a mean and standard deviation with transparent z-score conversion.
Translate z-scores into probability areas without a printed z table
For left-tail, right-tail and between-values questions, Lynara standardizes the entered boundary with z = (x − μ) ÷ σ and evaluates a bounded numerical approximation to the normal cumulative distribution function. The result is a model probability under the normal-distribution assumption, not a claim that the source data are actually normal.
Percentile mode solves the same relationship in reverse: it finds the cutoff with the requested cumulative area and converts that standardized z value back to the original scale. Keep empirical percentile rank separate when you are ranking observed raw data rather than assuming a normal model.
Privacy: calculator inputs are processed in your browser; Lynara does not need to upload these values to calculate the result.
Formula
z = (x − μ) ÷ σ; P(X ≤ x) = Φ(z); between probability = Φ(z₂) − Φ(z₁); inverse x = μ + zσ
Example
For μ = 0 and σ = 1, P(X ≤ 1.96) is about 0.9750; the 90th percentile is z ≈ 1.2816.
References for this method
Useful for
- Normal probability homework
- Left- and right-tail probabilities
- Between-values probability
- Percentile and inverse-normal calculations
Questions
Is this different from the Z-Score Calculator?
Yes. A z-score only standardizes a value; this page also converts normal-model z-scores into probability areas and works backward from a percentile to a raw value.
Can it find a percentile cutoff?
Yes. Choose percentile mode and enter a cumulative percentage strictly between 0% and 100% to solve for the corresponding x value.