Normal Distribution Calculators

Probability → boundary

Inverse Normal Distribution Calculator

Find a z-score or value from a percentile or probability. Choose left, right, or central area.

  • 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.

Quick examples

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.

Choose your area

Which inverse normal mode should I use?

Percentile or area below a cutoff

Choose Left tail. The 97.5th percentile means 97.5% of values lie below the cutoff: p = 0.975 in Decimal mode or 97.5 in Percent mode.

Top percentage or area above a cutoff

Choose Right tail. If only 5% of values exceed the cutoff, enter 0.05 in Decimal mode or 5 in Percent mode. For μ = 0 and σ = 1, the cutoff is z ≈ 1.64485363.

Middle percentage around the mean

Choose Central. A middle 95% area returns two symmetric boundaries and leaves 2.5% in each outside tail. On the standard normal scale, the bounds are approximately −1.95996398 and 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

Input checks

Common inverse normal input mistakes

  • Use 0.975 in Decimal mode or 97.5 in Percent mode; entering 97.5 as a decimal is invalid.
  • Enter standard deviation σ, not variance σ². If variance is 225, the standard deviation is √225 = 15.
  • A central 95% area is different from a left-tail 95th percentile: central mode returns two bounds; left-tail mode returns one.
  • Keep probability strictly between 0 and 1 (or 0% and 100%) and standard deviation greater than 0.

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.

Is a central normal interval a confidence interval for a mean?

Central mode gives an interval containing the selected proportion of values in the specified normal distribution. A confidence interval for an estimated mean also needs a sample size and an appropriate standard error or t model. This calculator does not estimate that interval from sample data.

Is InvNorm the same as InvT?

No. InvNorm uses the normal distribution; InvT uses Student’s t distribution and requires degrees of freedom. A question that specifies a t model needs an inverse t calculation, not this normal calculator.

Review and sources

Definitions and numerical references

Definitions and examples reviewed October 4, 2026

The formulas and example values are checked against numerical reference cases, including an independent high-precision oracle. Results use floating-point arithmetic and are rounded for display. Unsupported extreme inputs or overflow produce an error instead of an unverified answer.

The NIST table lists area between 0 and z; add or subtract from 0.5 when checking a left-tail probability. R and Python document the normal quantile operation used in the examples.

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.