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 Φ(z), the standard normal CDF, using the Abramowitz and Stegun 7.1.26 polynomial approximation, which is accurate to about 7 decimal places across the practical z range. Left-tail probability is Φ(z); right-tail is 1 − Φ(z); between two z-scores is Φ(z₂) − Φ(z₁).

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.

The result is exact to about 7 significant figures. The Abramowitz and Stegun approximation is the same one used by the p-value calculator on this site and by most introductory statistics software for closed-form CDF approximations. For research-grade precision (10+ decimals), use a statistical package or library that integrates the density directly.