Normal Distribution Calculator

The normal distribution is a symmetric, bell-shaped distribution defined by two parameters: mean μ and standard deviation σ. It is the most-used distribution in statistics because many natural processes are approximately normal, and because the Central Limit Theorem makes sums and averages of many independent random variables tend toward a normal distribution. The standard normal has μ = 0 and σ = 1.

The calculator first converts the raw score x to a z-score using z = (x − μ) ÷ σ. Then it computes the area under the standard normal curve using Hart's rational approximation. Each tail is calculated directly rather than by subtracting the other from 1, because subtracting a number very close to 1 destroys the accuracy of a small tail. That is why the left-tail and right-tail results agree exactly for values of equal magnitude, and why probabilities far out in the tail stay accurate instead of rounding to zero.

Use raw score mode when you have the actual values from your data, plus the mean and standard deviation of the underlying distribution. Use z-score mode when you already have the z-score (a standardized value) and just want the corresponding probability. The math is the same; the modes just save you the conversion step in one direction or the other.

Between mode computes P(a ≤ X ≤ b), the area under the curve strictly between two values. It is the difference between two CDFs: Φ(z₂) − Φ(z₁). For example, the probability that a value falls within 1 standard deviation of the mean (between z = −1 and z = +1) is about 0.6827, the well-known 68 percent rule for the normal distribution.

Φ(0) = 0.5 (the mean). About 68.27 percent of the area lies within z = ±1, 95.45 percent within z = ±2, and 99.73 percent within z = ±3. Common critical values: z = ±1.645 (90 percent two-sided), ±1.96 (95 percent), ±2.576 (99 percent). The calculator returns these directly when those z-scores are entered.

It is an approximation, but a close one. Checked against SciPy's normal distribution functions, the underlying calculation is correct to at least 8 significant figures everywhere tested, and to roughly 13 significant figures for z-scores within ±4 — the range that covers every conventional significance threshold. The same implementation is shared with the p-value calculator on this site, so the two always agree. Ordinary results are shown to six decimal places. A probability too small to survive that rounding switches to scientific notation instead — P(Z ≥ 8.3) reads 5.2056e-17 rather than 0.000000 — so a real value is never displayed as zero. A displayed zero therefore means a genuine zero: the probability has fallen below the smallest number the browser can represent, which happens past roughly z = ±38.6.