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Inverse Normal Calculator
The inverse normal calculator solves the reverse of the usual normal problem: given a probability, what value of x cuts off that area? Enter the mean, standard deviation and the probability, and say whether the area is to the left (a percentile), to the right (a top-p cut-off) or in the middle (symmetric around the mean). The steps convert the area to a left-tail area, find the z value with the inverse standard normal function, and unstandardise with x = μ + zσ.
Common questions it answers: what score marks the 90th percentile on a test with mean 500 and SD 100 (628.2); what IQ places someone in the top 2% (130.8); which two values enclose the middle 95% of a distribution (μ ± 1.96σ). This is the same calculation as invNorm on a TI-84 or NORM.INV in Excel.
The z values it produces are the critical values used in confidence intervals and z-tests, so the calculator doubles as a critical value finder. Go the other way, from a value to an area, with the normal distribution calculator.
Result and step-by-step solution
z = 1.2816
- Area to the leftP(X ≤ x) = 0.9
- Inverse standard normalz = Φ⁻¹(0.9) = 1.2816
- Unstandardisex = μ + zσ = 500 + (1.2816) × 100 = 628.1552
Frequently asked questions
- How do I find the value at the 90th percentile?
- Choose “to the left” with p = 0.90. The z value is 1.2816, so x = μ + 1.2816σ.
- How do I find the cut-off for the top 5%?
- Choose “to the right” with p = 0.05. The left-tail area is 0.95, z = 1.645, x = μ + 1.645σ.
- What is the middle 95% option?
- It finds the two values that enclose 95% of the distribution, leaving 2.5% in each tail: μ ± 1.96σ.
- Does this match invNorm on the TI-84?
- Yes. invNorm(area, μ, σ) with the area to the left gives the same result as the “to the left” option.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.