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.
At a glance
- Computes
- Normal-curve area below, above, between, or outside your values, and the inverse lookup.
- You supply
- Mean, standard deviation, and either a value x or a probability p.
- Use when
- Your quantity is normally distributed and you need a probability or a cutoff.
- Not for
- Small samples with an unknown population SD. Critical Value Calculator
Probability density; N(0, 1)
Probability
Enter a value to see its area
Set μ, σ, and a bound, then calculate.
| Detail | Value |
|---|
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).
Tail areas use the complementary incomplete-gamma form Q(1/2, z²/2), equivalent to erfc, so a small finite tail is calculated directly instead of by subtracting a rounded CDF from 1. The display keeps the page's conservative 1e-7 probability floor: values below it use a strict bound while the export records the numerical estimate and its precision state. The inverse uses Acklam's rational approximation (about 1e-9). To summarize a data set before assuming normality, start from descriptive statistics.
The chart requires a finite, ordered actual-x range. Rescale values that exceed drawable precision.
Everything on this page is continuous. A count is different. For successes in a fixed number of trials, the binomial distribution page computes each exact count probability. For events in an interval at a known rate, the Poisson distribution page does. Both print a normal approximation beside the exact answer and state the convention used to judge it.
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 .