Inverse Normal Distribution Calculator

Work out inverse normal distribution instantly with clear inputs, formula shown and shareable results.

About Inverse Normal Distribution Calculator

The inverse normal, or quantile function, answers the reverse question: which value has a given proportion of the distribution below it. Finding z first and then scaling by σ and shifting by μ gives the answer.

Formula

Inverse normal

x = μ + σ·Φ⁻¹(p)

Worked example

With the default values below, Inverse Normal Distribution Calculator returns Value with that cumulative probability of 129.39945979. Change any input above and every figure here is recalculated in your browser.

Inputs used

Cumulative probabilityaccepts 0.01–99.99 %: 97.5 %
Distribution mean: 100
Distribution standard deviationminimum 0: 15

Results

Value with that cumulative probability: 129.39945979
Corresponding z score: 1.95996399
Cumulative probability at that z: 0.97500007
Distance from the mean: 29.39945979

Step by step

1Values used
2Inverse normal
3Value with that cumulative probability= 129.39945979
4Corresponding z score= 1.95996399
5Cumulative probability at that z= 0.97500007
6Distance from the mean= 29.39945979

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For p = 97.5% with μ = 100, σ = 15?

z = 1.96, so x = 100 + 29.4 = 129.4 — the classic IQ threshold for the top 2.5%.

Why is there no closed form?

Φ has no elementary inverse, so rational approximations such as Acklam's are used, accurate to about 10⁻⁹.