Inverse Normal Calculator (percentile → value)

Understanding the inverse normal calculator — concept and principle

An inverse normal calculator works backwards from a probability to a value. You enter a percentile (the area to the left, in percent) together with a mean and standard deviation, and it returns the value x that cuts off that area. People use it to find cut-off scores ("what score is the 90th percentile?"), tolerance limits, and the critical z-values used in confidence intervals and hypothesis tests.

Φ⁻¹ turns 95% into z = 1.6449; a two-sided 95% leaves 2.5% per tail and needs 1.96.

How it is calculated

z = Φ⁻¹(p), where p = percentile ÷ 100 and Φ⁻¹ is the inverse of the standard normal CDF (also called the quantile or probit function).
x = μ + σ · z, which rescales the standard value to your distribution.
Two-sided z for this confidence = Φ⁻¹(1 − (1 − p) ÷ 2). This treats your percentage as a confidence level C and splits the remaining 1 − C equally between both tails.

The percentile must be strictly between 0 and 100; at 0 or 100 the answer would be −∞ or +∞. μ and x share the same unit, and σ must be positive.

The principle

Φ(z) rises smoothly from 0 to 1, so every probability between 0 and 1 corresponds to exactly one z. Φ⁻¹ has no closed-form expression, so it is computed numerically. The calculator uses Peter Acklam's algorithm, which approximates Φ⁻¹ with rational functions (one formula for the central region, another for the tails) and then applies one refinement step to polish the result. Because all normal curves have the same shape, a single standard quantile z can be moved to any distribution with x = μ + σz.

Worked example

Inputs: percentile = 95%, mean = 0, SD = 1 (the standard normal).
p = 95 ÷ 100 = 0.95, so z = Φ⁻¹(0.95) ≈ 1.6449.
x = 0 + 1 × 1.6449 ≈ 1.6449.
As a confidence level, each tail holds (1 − 0.95) ÷ 2 = 0.025, so the two-sided z = Φ⁻¹(0.975) ≈ 1.9600.

Each point is the calculator re-run with only “SD” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

SD	x
0.25	0.4112134067
0.5	0.8224268135
0.75	1.23364022
1	1.644853627
1.25	2.056067034
1.5	2.46728044
2	3.289707254

How to read the result

Common reference values: Φ⁻¹(0.90) ≈ 1.2816, Φ⁻¹(0.95) ≈ 1.6449, Φ⁻¹(0.975) ≈ 1.9600, Φ⁻¹(0.99) ≈ 2.3263 and Φ⁻¹(0.995) ≈ 2.5758. A percentile below 50% gives a negative z and an x below the mean. With μ = 100 and σ = 15, the 95th percentile is 100 + 15 × 1.6449 ≈ 124.7.

Common mistakes

Using the one-sided value 1.645 for a 95% two-sided interval; that needs 1.960.
Entering 0.95 when the field expects a percentage (95).
Using the normal quantile for small samples with unknown σ, where the t distribution is appropriate.

Keep in mind: The returned value is meaningful only if the variable is approximately normal. For skewed data, use empirical percentiles from the data instead.

Frequently asked questions

Why does 95% give 1.645 but 95% confidence uses 1.96?

1.645 leaves 5% in one tail. A two-sided 95% interval leaves 2.5% in each tail, so it needs Φ⁻¹(0.975) = 1.96.

What is the z for the 50th percentile?

Exactly 0, so x equals the mean.

Can I find the value for the top 10%?

Yes. The top 10% starts at the 90th percentile, so enter 90.

Sources: Acklam, P. J., An algorithm for computing the inverse normal cumulative distribution function (2003) · NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

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