Inverse Normal Distribution Calculator
Find the quantile for a normal distribution from a cumulative probability. Enter p as a decimal (0.95) or percent (95).
What is an inverse normal distribution?
The inverse normal distribution (also called the normal quantile function or probit) answers this question: “Given a probability, what value of x corresponds to that cumulative area under the normal curve?”
If the cumulative distribution function is written as Φ(x), then the inverse is Φ-1(p). For a standard normal variable (mean 0, standard deviation 1), the result is a z-score. For any normal distribution with mean μ and standard deviation σ:
x = μ + σ · Φ-1(p)
How to use this calculator
- Enter a cumulative probability p between 0 and 1 (or as 0–100 percent).
- Set μ (mean) and σ (standard deviation).
- Click Calculate Quantile.
- Read both outputs:
- z = standard normal quantile
- x = quantile for your N(μ, σ²) distribution
Example calculations
Example 1: Standard normal, 95th percentile
Input p = 0.95, μ = 0, σ = 1. The calculator returns z ≈ 1.64485. That means 95% of values lie below 1.64485.
Example 2: Exam scores
Suppose scores are normally distributed with mean 70 and standard deviation 12. For p = 0.90, z ≈ 1.28155. Then x = 70 + 12(1.28155) ≈ 85.38. So the 90th percentile score is about 85.4.
Example 3: Lower-tail cutoffs
If p = 0.05, z ≈ -1.64485. Negative z-scores are expected in lower-tail thresholds. This is often used for one-sided hypothesis tests.
Common percentile and z-score values
| Percentile | Probability (p) | z = Φ-1(p) |
|---|---|---|
| 80th | 0.80 | 0.8416 |
| 90th | 0.90 | 1.2816 |
| 95th | 0.95 | 1.6449 |
| 97.5th | 0.975 | 1.9600 |
| 99th | 0.99 | 2.3263 |
Where inverse normal is used
- Statistics: critical values for confidence intervals and hypothesis tests.
- Quality control: setting tolerance limits and defect thresholds.
- Finance: Value at Risk models and tail risk cutoffs.
- Education: converting test percentiles into scaled scores.
- Data science: probability-to-score transforms and calibration workflows.
Tips for accurate results
- Probability must be strictly between 0 and 1 (not equal to 0 or 1).
- Use a positive standard deviation (σ > 0).
- If you enter 95, this calculator interprets it as 95% = 0.95.
- Very extreme probabilities (e.g., 0.999999) can produce very large absolute z-scores.
FAQ
Is this the same as a z-score calculator?
Partly. This tool starts with a probability and returns the z-score (or x-value), while many z-score calculators start with x and return probability.
What if my data are not normal?
Then inverse normal values may not represent true quantiles of your data. Consider empirical quantiles or a distribution that better matches your sample.
Why do I get an error at p = 1?
Because the normal quantile tends to +∞ at p = 1 and -∞ at p = 0. Finite numeric calculators require 0 < p < 1.
Bottom line
The inverse normal distribution calculator is the fastest way to move from a target probability to a cutoff value. Use it whenever you need percentiles, critical z-values, or quantiles for any normal distribution.