Inverse Normal Distribution Calculator — Probit / Quantile
Given a cumulative probability p, find the value x such that P(X ≤ x) = p for a normal distribution with the specified mean and standard deviation.
The value x such that P(X ≤ x) equals the input probability
How does this calculator work?
The inverse normal distribution (probit) finds the value x such that P(X ≤ x) = p for a normal distribution with given mean μ and standard deviation σ. The formula x = μ + σ·Φ⁻¹(p) uses a rational approximation accurate to 10 significant figures. Standard normal z-scores (μ=0, σ=1) are shown alongside x.
Formula
How this is calculated
The normal (Gaussian) CDF Φ(z) gives the probability that a standard normal variable is ≤ z. The inverse problem — given a probability p, find the corresponding z — is solved by the probit function Φ⁻¹(p). Because no closed-form formula exists, the calculator uses the rational polynomial approximation by Peter Acklam, which is accurate to better than 1.15 × 10⁻⁹ absolute error across the entire (0, 1) interval.
The result for a general normal distribution with mean μ and standard deviation σ is x = μ + σ · z. Setting μ = 0 and σ = 1 gives the pure z-score. Common uses: finding critical values for hypothesis tests (e.g. p = 0.975 gives z ≈ 1.96, the two-tailed 95% confidence-interval boundary), computing percentile cutoffs, and calibrating signal-detection models.
The function is only defined for p strictly between 0 and 1. At the boundaries (p = 0 or p = 1) the result is ±∞. Very small or very large probabilities require accurate tail approximations — this implementation handles tails below p = 0.02425 and above p = 0.97575 with a dedicated rational polynomial to maintain precision.
Frequently asked questions
The probit function (probability unit) is the quantile function of the standard normal distribution — the inverse of Φ(z). Probit(0.5) = 0 (the median), probit(0.975) ≈ 1.96 (the 97.5th percentile), and probit(0.9) ≈ 1.282.
For a two-tailed test at the 5% significance level, you need the 97.5th percentile (p = 0.975) of the standard normal. Enter p = 0.975 with μ = 0 and σ = 1 to get z ≈ 1.96 — the value that leaves 2.5% in each tail.
Yes — enter your distribution's mean and standard deviation in the corresponding fields. The calculator scales the z-score to x = μ + σ·z automatically, so you can work directly with the original variable units.
Also known as
TG we-Calculate Editorial Team. (2026). Inverse Normal Distribution Calculator — Probit / Quantile [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/inverse-normal-distribution-calculator
TG we-Calculate Editorial Team. "Inverse Normal Distribution Calculator — Probit / Quantile." TG we-Calculate. 2026. https://we-calculate.com/calculator/inverse-normal-distribution-calculator.
TG we-Calculate Editorial Team, "Inverse Normal Distribution Calculator — Probit / Quantile," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/inverse-normal-distribution-calculator
@misc{wecalculate_inverse_normal_distribution_calculator, title = {Inverse Normal Distribution Calculator — Probit / Quantile}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/inverse-normal-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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