What's the difference between the PDF and CDF results?

The probability density function (PDF) gives the relative likelihood of the distribution at exactly x — a height on the bell curve, not a probability itself. The cumulative distribution function (CDF), P(X ≤ x), gives the actual probability of observing a value at or below x.

How accurate is the CDF approximation?

The calculator uses the Abramowitz-Stegun polynomial approximation of the normal CDF, accurate to within about 7.5×10⁻⁸, more than precise enough for statistics coursework and most practical applications.

Normal Distribution Calculator

Calculate probability density and cumulative probability for a normal distribution.

How this is calculated

PDF(x) = (1 ÷ (σ√2π)) × e^(−(x−μ)² ÷ 2σ²); CDF via the Abramowitz-Stegun approximation

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 calculator finds the probability density at a point and the cumulative probability P(X ≤ x) for any normal distribution, plus the probability of falling within a range.

Worked example

N(100, 15) at x=115

Result: P(X≤115) ≈ 84.13%

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Frequently asked questions

Z-Score & P-Value Calculator

Calculate a z-score, percentile, and p-value from a value, mean and standard deviation.

Normal Distribution Calculator

About this calculator

Frequently asked questions

What's the difference between the PDF and CDF results?

How accurate is the CDF approximation?

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