Normal Distribution Calculators

Probability → boundary

Inverse Normal Distribution Calculator

Enter a percentile or probability to find the matching z-score and value for any mean and standard deviation.

  • Runs locally in your browser
  • No sign-up
  • Shows z and raw x
Probability area
Probability format

Enter a decimal greater than 0 and less than 1.

Any finite number.

Must be greater than 0.

Ready example

97.5th percentile

Use the prefilled values, then choose Calculate inverse normal.

-4-2024z = 1.95996
Curve readingExample preview: a left-tail probability of 0.975 ends at z ≈ 1.95996398.

Method

How inverse normal calculation works

  1. Convert the selected area to a left-tail probability

    The inverse normal function uses cumulative area from the left. A left-tail input is already in that form. For a right-tail input q, the corresponding left-tail area is 1 − q. For a central area c, each outside tail is (1 − c) / 2.

  2. Find the standard normal boundary

    The calculator evaluates z = Φ⁻¹(p), where Φ is the standard normal cumulative distribution function. Central mode evaluates the lower and upper tail boundaries separately so both endpoints remain numerically stable.

  3. Transform z to the requested scale

    For a normal distribution with mean μ and standard deviation σ, the raw value is x = μ + σz. The calculator reports both z and x so the standardized and original-scale answers stay visible together.

Checked examples

Worked inverse normal examples

Standard normal 97.5th percentile

Left-tail p = 0.975, μ = 0, σ = 1

z ≈ 1.95996398 and x ≈ 1.95996398

A score on a custom scale

Left-tail p = 0.975, μ = 100, σ = 15

z ≈ 1.95996398 and x ≈ 129.39945977

Central 95% interval

Central probability = 0.95, μ = 0, σ = 1

lower z ≈ −1.95996398 and upper z ≈ 1.95996398

FAQ

Inverse normal questions

What is an inverse normal distribution calculation?

It starts with a cumulative probability or percentile and returns the boundary value that leaves that amount of normal-curve area on the selected side. This is the reverse direction of calculating probability from a known x-value or z-score.

Is InvNorm the same as the inverse normal CDF?

Usually, yes. Calculator labels such as InvNorm commonly refer to the inverse cumulative distribution function. Always check whether the tool expects a left-tail area, because right-tail and central-area questions require a conversion first.

Can probability be exactly 0 or 1?

No finite normal quantile exists at exactly 0 or 1. The normal curve extends indefinitely, so those endpoints correspond to negative or positive infinity. This calculator requires a probability strictly between 0 and 1, or between 0% and 100%.

How do I convert a percentile to a z-score?

Write the percentile as a decimal left-tail probability, then evaluate Φ⁻¹(p). For example, the 97.5th percentile is p = 0.975 and gives z ≈ 1.95996398. Percent mode performs the division by 100 without changing the probability meaning.

Review and sources

Definitions and numerical references

Reviewed July 13, 2026

This educational calculator is designed for learning and routine checking. Verify medical, financial, legal, engineering-safety, or other high-stakes work independently.

Already have x or z? Calculate the area under the normal curve.