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Inverse Normal Distribution Calculator

Free inverse normal distribution calculator. Find quantiles and percentiles given probability. Left

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

Φ⁻¹
STATISTICSInverse CDF / Probit

Find x-Value (Quantile) Given a Probability

Probit function. Percentiles, VaR, tolerance intervals. Left tail, right tail, two-tailed symmetric.

Normal Distribution →Z-Score →

Real-World Scenarios — Click to Load

Tail Direction

Inputs

Normal Curve with Shaded Tail Area

inverse_normal.sh
CALCULATED
$ inverse_normal --p=0.975 --mean=0 --std=1 --tail="left"
P(X ≤ x) = 97.50%
x = 1.9600
Z-Score
1.9600
Percentile
97.50%
Share:
Inverse Normal Distribution
P(X ≤ x) = 97.50%
1.9600
z = 1.96μ = 0σ = 1
numbervibe.com/calculators/statistics/inverse-normal-distribution-calculator

Probit Function Φ⁻¹(p)

Calculation Breakdown

COMPUTATION
Probit Φ⁻¹(p)
1.9600
ext{Standard} ext{normal} ext{quantile}
x = μ + σ·Φ⁻¹(p)
1.9600
x = μ + σ·Φ⁻¹(p) = 0 + 1·Φ⁻¹(0.9750) = 1.9600
RESULT
Z-Score
1.9600
z = (x - \text{mu} ) / \text{sigma}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • The inverse normal CDF (probit) finds the x-value corresponding to a given cumulative probability
  • Left tail: P(X ≤ x) = p → x = μ + σ·Φ⁻¹(p). Right tail: P(X ≥ x) = p → use Φ⁻¹(1-p)
  • For symmetric intervals: P(a ≤ X ≤ b) = p → a = Φ⁻¹((1-p)/2), b = Φ⁻¹((1+p)/2)
  • Applications: percentiles, tolerance intervals, Value at Risk (VaR), quality control limits
  • The probit function Φ⁻¹(p) is the inverse of the standard normal CDF

Did You Know?

📊The 95th percentile of the standard normal is z ≈ 1.645 — used in one-sided confidence intervalsSource: NIST
🧠IQ scores: "Top 2%" means P(X≥x)=0.02, so x = 100 + 15×Φ⁻¹(0.98) ≈ 131Source: APA
📈Value at Risk (VaR) at 95% uses the 5th percentile — the loss level exceeded 5% of the timeSource: Basel III
🏭Quality control uses inverse normal for tolerance intervals — e.g., 99.7% of parts within ±3σSource: ISO
🎓GPA percentiles: "Top 5%" means finding the GPA such that 95% of students score below itSource: ETS
🌡️Temperature ranges: "Middle 90%" gives symmetric bounds — 5% below lower, 5% above upperSource: Meteorology

How It Works

1. The Probit Function

Φ⁻¹(p) is the inverse of the standard normal CDF. Given probability p, it returns the z-score. No closed form — computed via rational approximation (Beasley-Springer-Moro).

2. Left Tail (Percentile)

P(X ≤ x) = p. Direct: x = μ + σ·Φ⁻¹(p). Example: 95th percentile of N(0,1) is Φ⁻¹(0.95) ≈ 1.645.

3. Right Tail

P(X ≥ x) = p means P(X ≤ x) = 1-p. So x = μ + σ·Φ⁻¹(1-p). Example: Top 2% means p=0.02, use Φ⁻¹(0.98).

4. Two-Tailed Symmetric

P(a ≤ X ≤ b) = p, symmetric around μ. Lower tail has (1-p)/2, upper has (1+p)/2.

5. Value at Risk (VaR)

VaR at confidence 1-α is the (1-α)th percentile of the loss distribution. VaR 95% uses the 5th percentile (left tail).

Expert Tips

Left vs Right

Left: P(X≤x)=p. Right: P(X≥x)=p → convert to P(X≤x)=1-p

Common Percentiles

90%: z≈1.28, 95%: z≈1.645, 97.5%: z≈1.96, 99%: z≈2.33

Tolerance Intervals

99.7% symmetric → ±3σ. 95% symmetric → ±1.96σ

Excel / R

Excel: NORM.INV(p,μ,σ). R: qnorm(p, mean, sd)

Forward vs Inverse

DirectionForward (CDF)Inverse (Probit)
Givenxp (probability)
Findp = P(X≤x)x (quantile)
FunctionΦ((x-μ)/σ)Φ⁻¹(p)·σ + μ
Use caseWhat % below x?What x gives p%?

Frequently Asked Questions

What is the probit function?

The probit is Φ⁻¹(p), the inverse of the standard normal CDF. Given a probability p, it returns the z-score such that P(Z ≤ z) = p.

How do I find the 95th percentile?

Use left tail with p=0.95. For N(0,1), x = Φ⁻¹(0.95) ≈ 1.645. For N(μ,σ), x = μ + 1.645σ.

What is Value at Risk (VaR)?

VaR at 95% confidence is the 5th percentile of the loss distribution — the loss level exceeded only 5% of the time. Use left tail with p=0.05.

How do I get symmetric bounds for "middle 90%"?

Use two-tailed with p=0.90. The calculator finds a and b such that P(a≤X≤b)=0.90, with 5% in each tail.

Why does "top 2%" use p=0.98 for the inverse?

Top 2% means P(X≥x)=0.02. So P(X≤x)=0.98. The inverse gives x such that 98% are below — that is the cutoff for top 2%.

What is the difference between percentile and quantile?

They are equivalent. The p-th quantile is the value below which p×100% of the distribution lies. The 95th percentile = 0.95 quantile.

Common Z-Scores (Standard Normal)

1.28
90th percentile
1.645
95th percentile
1.96
97.5th (95% CI)
2.33
99th percentile

Disclaimer: This calculator uses the Peter Acklam / Beasley-Springer-Moro algorithm for the inverse normal CDF. For critical applications (financial VaR, clinical thresholds), verify against established statistical software. Educational and professional reference.

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