Normal Distribution Calculator: Curve, Area, and Inverse
Draw a normal distribution curve and read the area below, above, between, or outside your values, with an inverse mode that finds x from a probability or percentile.
Probability
Enter a value to see its area
Set μ, σ, and a bound, then calculate.
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z-scores standardize each bound against N(0, 1). See the z-score calculator for the standardization step on its own.
Φ(z) = 1/2 · [1 + erf(z / √2)], z = (x - μ) / σ, x = μ + σ · Φ⁻¹(p)
How?
How this is calculated
Areas come from the cumulative distribution function Φ evaluated on the z-score z = (x - μ) / σ. Below is Φ(z), above is 1 - Φ(z), between is Φ(z_b) - Φ(z_a), and outside is 1 - [Φ(z_b) - Φ(z_a)]. The inverse solves x = μ + σ · Φ⁻¹(p) for a left-tail probability p in (0, 1).
Φ uses the Abramowitz and Stegun 7.1.26 approximation to erf. Its error is bounded in absolute terms (about seven correct decimal places), not in significant figures, so deep-tail areas from the above and outside modes are shown clamped to that floor rather than padded with digits the method cannot support. The inverse uses Acklam's rational approximation (about 1e-9). To summarize a data set before assuming normality, start from descriptive statistics.
Formula: Φ(z) = 1/2 · [1 + erf(z / √2)], z = (x - μ) / σ, x = μ + σ · Φ⁻¹(p)
The empirical rule (68-95-99.7)
For a normal distribution, the between mode returns the areas the empirical rule names: about 68.2689% of the area lies within ±1σ of the mean, 95.4499% within ±2σ, and 99.7300% within ±3σ. These are the exact values this tool computes for N(0, 1) between -1 and 1, -2 and 2, and -3 and 3.
Sources
- NIST SEMATECH e-Handbook, Normal distribution (1.3.6.6.1). NIST. Retrieved .
- Abramowitz and Stegun, Handbook of Mathematical Functions, 7.1.26 (p. 299). National Bureau of Standards. Retrieved .
- Peter Acklam, An algorithm for computing the inverse normal cumulative distribution function. Peter John Acklam. Retrieved .