Normal Distribution Probability Calculator
Find the probability (area) under any normal curve for tail or interval queries, or work backwards from a probability to the value of x.
Mode
90.8789% of the distribution
- 1
Standardize x
z = (120 − 100) ÷ 15 = 1.3333 - 2
P(X < x) = Φ(z)
Φ(1.3333) = 0.908789Φ(z) is the standard normal cumulative distribution function.
How does this calculator work?
Enter the mean and standard deviation, pick a mode, and the calculator standardizes your value to a z-score and returns the area under the normal curve: P(X<x), P(X>x), or the area between two points. Inverse mode turns a probability back into the matching x value.
Formula
How this is calculated
You describe the bell curve with its mean μ (the center) and standard deviation σ (the spread, which must be positive). Each value x is standardized into a z-score, z = (x − μ) / σ, that measures how many standard deviations x sits from the mean. The standard normal cumulative function Φ(z) = ½(1 + erf(z/√2)) gives the area to the left of z, and the error function erf is evaluated with the Abramowitz–Stegun rational approximation (accurate to about 1.5e-7).
The mode selects the query. "Left" returns P(X < x) = Φ(z). "Right" returns P(X > x) = 1 − Φ(z). "Between" standardizes both boundaries and returns Φ(z₂) − Φ(z₁); the two x values are sorted automatically so order does not matter. "Inverse" reverses the process: given a cumulative probability p between 0 and 1, it finds the z-score with Acklam's rational approximation for the inverse normal CDF, then maps it back with x = μ + σ·Φ⁻¹(p).
Probabilities are unitless areas between 0 and 1 (shown also as a percentage); x is in whatever units μ and σ use. Edge cases are guarded: σ must be greater than zero, and the inverse mode requires 0 < p < 1 since the tails extend to ±∞. The plotted curve is the standard normal shape from μ−4σ to μ+4σ for visual reference.
Frequently asked questions
A z-score is the number of standard deviations a value lies from the mean, z = (x − μ) / σ. It converts any normal distribution to the standard normal so a single table or function can give the area.
Φ(z) is the cumulative area, which ranges strictly between 0 and 1. A probability of exactly 0 or 1 would correspond to z = ±∞, so the inverse is only defined for 0 < p < 1.
The CDF uses the Abramowitz–Stegun erf approximation (error ≈ 1.5×10⁻⁷) and the inverse uses Acklam's algorithm (error ≈ 1.15×10⁻⁹), both far tighter than printed z-tables.
Also known as
TG we-Calculate Editorial Team. (2026). Normal Distribution Probability Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/normal-distribution-probability-calculator
TG we-Calculate Editorial Team. "Normal Distribution Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/normal-distribution-probability-calculator.
TG we-Calculate Editorial Team, "Normal Distribution Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/normal-distribution-probability-calculator
@misc{wecalculate_normal_distribution_probability_calculator, title = {Normal Distribution Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/normal-distribution-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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