Normal Distribution Calculator
Calculate probability density and cumulative probability for a normal distribution.
Assumptions used in this calculation
- CDF approximation: The cumulative probability is computed via the Abramowitz-Stegun numerical approximation rather than an exact closed-form integral (none exists for the normal distribution), with error under 7.5×10⁻⁸.
About this calculator
The normal distribution's cumulative probability, the chance a value falls at or below a given point, has no exact closed-form formula, so computing P(X ≤ x) by hand normally means looking up a standardized z-table and interpolating, an extra conversion step for anything not exactly on the table's grid. This calculator takes a mean, standard deviation and value x for any normal distribution and computes both the probability density function, PDF(x) = (1 ÷ (σ√2π)) × e^(−(x−μ)² ÷ 2σ²), the height of the bell curve at exactly x (not itself a probability), and the cumulative distribution function, P(X ≤ x), the actual probability of observing a value at or below x, using the Abramowitz-Stegun polynomial approximation, accurate to within about 7.5×10⁻⁸. It also computes the probability of falling within a specified range by taking the difference of two CDF evaluations. That skips the z-table lookup and interpolation entirely, working directly with your distribution's actual mean and standard deviation instead of first standardizing to z-scores.
Worked example
N(100, 15) at x=115
Result: P(X≤115) ≈ 84.13%
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