Free calculator
Inverse normal distribution calculator
Enter a left-tail probability p with 0 < p < 1 (meaning P(X ≤ x) = p for X ~ N(μ, σ²)), plus mean μ and standard deviation σ > 0. You get the cutoff x and the standardized z = (x − μ) / σ. Toggle standard normal (μ = 0, σ = 1) for InvNorm / NORM.S.INV-style problems. For P(X ≤ x) or between probabilities on the same model, use this site’s normal distribution calculator—this page stays on inverse (quantile) math only.
When to use this calculator
Fast quantile checks before you mirror NORM.INV / NORM.S.INV in Google Sheets or Excel—transparent formulas, not a full statistics suite.
- Turn a cumulative left-tail probability into a cutoff x for a stated μ and σ (percentiles, simple critical values, intro stats).
- Match
NORM.INV(p, μ, σ)orNORM.S.INV(p)cells against this page when auditing a template row. - Need P(X ≤ x) or P(a ≤ X ≤ b) on N(μ, σ²)? Use the normal distribution calculator; need (x − μ)/σ with tail readouts for one x? Use the z-score calculator.
For X ~ N(μ, σ²) with σ > 0, the cumulative distribution is F(x) = P(X ≤ x). The inverse (quantile) answers: given p in (0,1), find x with F(x) = p—the same object as NORM.INV / NORM.S.INV in spreadsheets for that convention.
Quantile definition
The value x = F⁻¹(p) is the smallest threshold whose left-tail area reaches p. On a continuous normal model, that matches NORM.INV with cumulative semantics.
From z to x
Compute z = Φ⁻¹(p) on N(0,1), then x = μ + σ z. This is the same rearrangement as z = (x − μ)/σ read backward.
Right-tail or two-sided shortcuts
If a problem gives P(X > x) = q, use p = 1 − q here. For two-sided critical values tied to α, common textbook cutoffs use p = 1 − α/2 in the upper tail story—see the confidence interval calculator for interval wording.
This page does not estimate μ or σ from raw rows; paste summary values or use the standard deviation tool first when σ comes from data.
FAQs cover left vs right tail, InvNorm vs NORM.INV, percentiles, and calculator menus—without becoming exam software.
Google Sheets & Excel
In English function names, NORM.INV(p, mean, standard_dev) returns x with P(X ≤ x) = p for X ~ N(μ, σ). NORM.S.INV(p) is the same for Z ~ N(0,1) (so x = z). Replace cell references with your values. On localized Excel, use Insert function to find the same patterns under your language pack.
=NORM.INV(D2, B2, C2)D2 = p in (0,1), B2 = μ, C2 = σ (> 0).
=NORM.S.INV(D2)D2 = p for Z ~ N(0,1); result is both z and x when μ = 0 and σ = 1.
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Frequently asked questions
What is an inverse normal calculator?
It finds a cutoff x on a normal model N(μ, σ²) from a left-tail probability p with P(X ≤ x) = p. It is the same quantile idea as NORM.INV / NORM.S.INV in Google Sheets and Excel (English names).
What does “left-tail p” mean here?
p is always P(X ≤ x) on the stated N(μ, σ²). That matches NORM.INV(p, μ, σ) and typical InvNorm “area to the left” defaults—not a right-tail area unless you transform p first.
My problem gives a right-tail area. What p do I enter?
If P(X > x) = q, then P(X ≤ x) = 1 − q, so enter p = 1 − q in this tool (with 0 < p < 1).
How do I get a two-sided critical z for confidence α?
For the common symmetric two-sided normal critical value, textbooks use the upper cutoff z with P(Z > z) = α/2, which corresponds to P(Z ≤ z) = 1 − α/2. You can enter p = 1 − α/2 here with μ = 0 and σ = 1—then x equals that z. For full interval math on means/proportions, use the confidence interval calculator.
How is this different from your normal distribution calculator?
This page is only p → x (plus z) with a streamlined layout. The normal distribution calculator also computes P(X ≤ x), P(X ≥ x), and between probabilities on N(μ, σ²) in one tool.
Is p the same as a percentile?
A percentile is often reported as 100 × p for the same left-tail p. For example p = 0.90 corresponds to the 90th percentile cutoff under left-tail semantics used here and in NORM.INV.
What are the Google Sheets / Excel formulas?
=NORM.INV(p, mean, standard_dev) returns x with P(X ≤ x) = p for X ~ N(μ, σ). =NORM.S.INV(p) is the same for Z ~ N(0,1). Use TRUE cumulative forms when you combine with NORM.DIST / NORM.S.DIST on the forward direction.
Which Excel German names match this page?
Use Formulas → Insert function to confirm names in your install. Patterns map as NORM.INV ↔ NORMINV / NORM.INV, NORM.S.INV ↔ NORM.S.INV in many German installs—always verify against your language pack.
Which Excel French names match this page?
Use Formules → Insérer une fonction for locale-accurate names. Common pairs include NORM.INV with LOI.NORMALE.INVERSE and NORM.S.INV with LOI.NORMALE.STANDARD.INVERSE—verify in your workbook.
Does this replace my Casio / TI InvNorm menu?
No—menus differ by model and syllabus. Use this page to check the same p, μ, and σ story, then follow your course’s calculator steps for exams.
Will this match Excel exactly?
Results follow the same double-precision normal model used across this site’s statistics tools; tiny differences vs a spreadsheet can still appear at the last decimal due to formatting—compare enough digits when it matters.
Is this statistical advice?
No—educational and illustrative only. Pick methods with your instructor or team when decisions depend on the numbers.