Inverse Normal Distribution Calculator

Find the value x that a chosen cumulative probability falls below on a normal distribution. Enter a probability plus the mean and standard deviation to get the quantile x = μ + σ·z.

Quick Facts

Method
Inverse CDF (probit): x = μ + σ·Φ⁻¹(p)
z is computed with Acklam's rational approximation of the standard-normal quantile (error < 1.2×10⁻⁹).

Your Results

Calculated
Quantile value (x)
-
The value with p of the distribution below it
Z-score (standard-normal quantile)
-
Φ⁻¹(p), before applying μ and σ
Percentile
-
p expressed as a percentage
Upper-tail probability
-
1 − p (fraction of values above x)

Ready

Enter a probability, mean, and standard deviation, then calculate.

What the inverse normal distribution finds

A normal (Gaussian) distribution is fully described by two numbers: its mean μ (the center) and its standard deviation σ (the spread). The ordinary cumulative distribution function (CDF), written Φ, answers "what fraction of values fall at or below x?" The inverse normal distribution reverses that question: given a target cumulative probability p, it returns the value x that has exactly that fraction of the distribution below it. That value is called a quantile or percentile value.

For example, if adult resting heart rates are roughly normal with μ = 72 bpm and σ = 9 bpm, and you want the 95th-percentile heart rate, you ask the inverse: "what value has 0.95 of the distribution below it?" The calculator returns x ≈ 86.8 bpm.

The formula

The calculation has two steps. First find the standard-normal quantile (also called the z-score or probit) for probability p — the value on a standard normal curve (μ = 0, σ = 1) with p below it:

z = Φ⁻¹(p)

Then rescale and shift it to your distribution:

x = μ + σ · z = μ + σ · Φ⁻¹(p)

There is no elementary closed-form expression for Φ⁻¹, so this tool evaluates it with Acklam's rational approximation, a widely used algorithm whose maximum absolute error is below 1.2 × 10⁻⁹ across the whole (0, 1) interval — far more precise than any statistical table.

Common z-values worth memorizing

  • p = 0.50 → z = 0 (the median equals the mean for a normal distribution)
  • p = 0.90 → z ≈ 1.2816
  • p = 0.95 → z ≈ 1.6449 (one-sided 95% critical value)
  • p = 0.975 → z ≈ 1.9600 (the "1.96" behind a 95% two-sided confidence interval)
  • p = 0.99 → z ≈ 2.3263
  • p = 0.995 → z ≈ 2.5758 (used for two-sided 99% intervals)

Because the standard normal is symmetric, Φ⁻¹(p) = −Φ⁻¹(1 − p). For instance the 5th percentile z is −1.6449, the mirror image of the 95th.

Where this calculation is used

  • Confidence intervals and critical values: the ±1.96 and ±2.576 multipliers come straight from Φ⁻¹(0.975) and Φ⁻¹(0.995).
  • Hypothesis testing: converting a chosen significance level α into a rejection cutoff.
  • Quality control and Six Sigma: finding process limits at a target yield or defect rate.
  • Finance: Value-at-Risk uses the quantile of a return distribution (e.g. the 1st percentile 1-day loss).
  • Standardized testing and percentiles: mapping a percentile rank back to a raw score.

Frequently Asked Questions

How is this different from a normal distribution (CDF) calculator?
A CDF calculator goes forward: you give it an x-value and it returns the probability p = P(X ≤ x). The inverse normal goes backward: you give it a probability p and it returns the x-value. They undo each other — if Φ(x) = p, then Φ⁻¹(p) = x.
Why does p = 0.5 return exactly the mean?
The normal distribution is symmetric about its mean, so exactly half its area lies below μ. The 50th percentile (median) therefore equals the mean, and the standard-normal z for p = 0.5 is 0.
What happens at p = 0 or p = 1?
The quantile is undefined there: the normal distribution extends to ±∞, so no finite value has 0% or 100% of the area below it. The calculator requires 0 < p < 1 and rejects the endpoints.
Where does the famous 1.96 come from?
A two-sided 95% interval leaves 2.5% in each tail, so the upper cutoff sits at p = 0.975. Φ⁻¹(0.975) ≈ 1.9600, which is why 95% confidence intervals use mean ± 1.96 standard errors.

Related calculators