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Inverse Normal CDF Calculator

Find the value at any given percentile of a normal distribution by entering the probability, mean, and standard deviation. Powered by the Acklam rational approximation.

The cumulative probability P(X ≤ x).

Result

1.2816

For a normal distribution with mean 0 and standard deviation 1, the value at the 90th percentile is 1.2816

Quick Answer

The inverse normal CDF calculator finds the value x at a given cumulative probability p in a normal distribution: x = mean + standard deviation × Φ⁻¹(p). Φ⁻¹(p) is the standard-normal quantile, computed using the Acklam rational approximation. For a distribution with mean 100 and standard deviation 15 (IQ-scale), the value at the 90th percentile (p = 0.90) is 100 + 15 × 1.2816 = 119.22. If you have a value and want the probability, use the Normal CDF Calculator instead.

How An Inverse Normal CDF Calculator Works, Step By Step

The inverse normal CDF calculator solves the reverse problem from the normal CDF. The forward direction asks: given a value x, what fraction of a normally distributed population falls at or below it? The inverse direction asks: given that fraction (the cumulative probability p), what value x corresponds to it? This calculator handles the inverse direction. The calculation finds the value at a specific percentile of any normal distribution defined by its mean and standard deviation. It is the tool to use when a problem gives you a probability and asks for the corresponding value, such as finding the score that separates the top 10 percent of a test distribution, or the dosage threshold below which 5 percent of a population falls. The standard-normal quantile function Φ⁻¹(p) has no simple closed-form algebraic solution, so the calculator uses Peter Acklam's rational approximation, a widely used algorithm that gives results accurate to better than 1.15 × 10⁻⁹ relative error across the full probability range.

Inverse Normal CDF: The Formula Behind Every Result

x = μ + σ × Φ⁻¹(p). Φ⁻¹(p) is calculated via Peter Acklam's rational approximation algorithm for the standard-normal quantile function.

  • General formula: x = μ + σ × Φ⁻¹(p).
  • μ (mu) = population mean, σ (sigma) = standard deviation, p = cumulative probability.
  • Φ⁻¹(p) = standard-normal quantile function (Acklam rational approximation).
  • Relative error < 1.15 × 10⁻⁹ across full (0, 1) probability range.

Getting Started With Inverse Normal CDF: Step By Step

Inputs

  • Cumulative probability p between 0 and 1 exclusive (e.g. 0.90 for 90th percentile).
  • Mean (centre) of the normal distribution.
  • Standard deviation (spread) of the distribution (must be positive).

Steps

  1. Enter the probability p (for a percentile, divide by 100: 90th percentile = 0.90).
  2. Enter the mean of the distribution.
  3. Enter the standard deviation.
  4. Read the value x from the output field.
  5. See the shaded area on the bell-curve graphic to confirm the queried percentile visually.
  6. To solve the reverse problem (value to probability), use the Normal CDF Calculator.

Inverse Normal CDF Example With Every Working Step Shown

IQ-style test with mean = 100, standard deviation = 15, target = 90th percentile (p = 0.90)

  1. Find standard-normal quantile: Φ⁻¹(0.90) = 1.2816 (via Acklam approximation).
  2. Apply formula: x = 100 + 15 × 1.2816.
  3. Calculate: x = 100 + 19.224 = 119.22.

119.22 score sits at the 90th percentile (90% of scores fall at or below 119.22).

When An Inverse Normal CDF Calculator Is Most Useful

Use this calculator when a statistics problem gives you a probability or percentile and asks for the corresponding value. Common applications include finding score cutoffs for academic percentile rankings, computing value-at-risk thresholds in finance (a specific percentile of a loss distribution), setting quality-control limits (the value below which a defined fraction of parts fall), and finding reference intervals in clinical testing (the 2.5th and 97.5th percentile values that define a 95 percent reference range).

For other math calculations, visit the math calculators hub, and check the glossary for definitions of common technical terms.

Assumptions

  • The underlying distribution is assumed to be normal (Gaussian).
  • The probability p must be strictly between 0 and 1; values of exactly 0 or 1 are undefined.
  • Standard deviation must be a positive number.
  • Results assume a continuous, symmetric normal distribution; discrete or skewed distributions require different methods.

Limitations

  • Does not handle distributions other than normal (Gaussian); for t-distributions or chi-square, use dedicated calculators.
  • Not suitable for finding quantiles of empirical or unknown distributions.
  • The Acklam approximation has a relative error below 1.15 × 10⁻⁹, which is sufficient for practical work but is not an exact analytical result.
  • Does not compute confidence intervals or hypothesis test critical values directly.

In Practice

The most common confusion is between the inverse CDF and the CDF itself. The CDF answers "I have a value; what is the probability?" The inverse CDF answers "I have a probability; what is the value?" If you type a probability into the wrong calculator, you get a probability out when you wanted a value, or a value out when you wanted a probability. Check the direction first: value to probability runs the forward (CDF) direction; probability to value runs the inverse direction handled by this tool. The bell-curve graphic on this page shades the area to the left of the result, which makes the direction visually obvious.

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Inverse Normal CDF Calculator: Frequently Asked Questions

What Is the Inverse Normal CDF?

The inverse normal CDF is a function that takes a cumulative probability p and returns the value x at which a normal distribution accumulates exactly that probability. It is the mathematical inverse of the normal CDF: where the CDF maps values to probabilities, the inverse maps probabilities to values. It is also called the quantile function, the percent-point function, or the probit function for the standard normal.

How Is This Different From the Normal CDF Calculator?

The Normal CDF Calculator takes a value x and returns the cumulative probability: the fraction of the distribution at or below x. This calculator does the reverse: it takes a probability p and returns the corresponding value x. Use the Normal CDF when you have a score and want a percentile; use this calculator when you have a percentile and want a score.

What Algorithm Does This Calculator Use for Φ⁻¹(p)?

This calculator uses Peter Acklam's rational approximation for the standard-normal quantile function. Acklam's algorithm divides the (0, 1) probability range into a central region and two tail regions, applying different rational polynomial approximations to each for maximum accuracy. The relative error is below 1.15 × 10⁻⁹ across the full range, making it more than accurate enough for any practical statistical application. The same approach underlies Excel's NORM.S.INV function.

What Does a Probability of 0.95 Mean in This Calculator?

A probability of 0.95 means the 95th percentile: the value below which 95 percent of the distribution falls. Entering p = 0.95 with mean = 0 and standard deviation = 1 (the standard normal) returns the z-score 1.645, the critical value used in one-sided 95 percent hypothesis tests and confidence intervals.

Can I Use This for a Standard Normal Distribution?

Yes. Enter mean = 0 and standard deviation = 1 to get the standard-normal quantile (z-score) for any probability. The result is Φ⁻¹(p) directly, without any scaling. For example, p = 0.975 returns z = 1.960, the critical value for a two-sided 95 percent confidence interval.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.