Normal Probability Calculator

The calculator converts your value to a standard z-score, then applies the Abramowitz-Stegun approximation to evaluate the CDF of the standard normal distribution.

Step 1 — Standardize:

$$z = \frac{x - \mu}{\sigma}$$

where $$\mu$$ is the mean and $$\sigma$$ is the standard deviation.

Step 2 — Approximate the CDF:

For $$z \geq 0$$, the Abramowitz-Stegun rational approximation uses the substitution $$t = \frac{1}{1 + 0.2316419 \cdot z}$$ and computes:

$$\Phi(z) \approx 1 - \phi(z) \cdot t(a_1 + t(a_2 + t(a_3 + t(a_4 + t \cdot a_5))))$$

where $$\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$$ is the PDF and the coefficients are $$a_1 = 0.3194$$, $$a_2 = -0.3566$$, $$a_3 = 1.7815$$, $$a_4 = -1.8213$$, $$a_5 = 1.3303$$.

For $$z < 0$$, we use symmetry: $$\Phi(z) = 1 - \Phi(-z)$$.

Step 3 — Derive other probabilities:

$$P(X > x) = 1 - \Phi(z)$$

The PDF gives the height of the density curve at the point $$x$$:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

This approximation achieves absolute error less than $$7.5 \times 10^{-8}$$, which is more than sufficient for practical applications.

The z-score tells you how many standard deviations your value lies from the mean. A z-score of 0 means the value equals the mean; positive z-scores indicate values above the mean.

The CDF (P(X ≤ x)) gives the probability that a randomly drawn observation falls at or below your value. For instance, a CDF of 0.975 means 97.5% of the distribution lies below that point.

The PDF is the probability density — it represents the relative likelihood of a value occurring at that exact point. It is highest at the mean and decreases symmetrically.

The upper-tail probability P(X > x) is the complement of the CDF and is widely used in hypothesis testing (one-tailed p-values).