Inverse Normal Distribution Calculator
Find the value x such that a normal random variable satisfies a target probability.
Tip: For the 95th percentile of a standard normal distribution, use p = 0.95, μ = 0, σ = 1.
What is an inverse normal distribution calculator?
A normal distribution inverse calculator does the reverse of a regular probability lookup. Instead of starting with a value and finding probability, you start with a probability and find the value. In statistics, this is often called a quantile, percentile, or inverse CDF calculation.
If you know the cumulative probability p, the mean μ, and standard deviation σ, this calculator returns the cutoff point x such that:
- Lower tail: P(X ≤ x) = p
- Upper tail: P(X ≥ x) = p
Core formula
For a normal distribution X ~ N(μ, σ²), the inverse relationship is:
x = μ + σ · zp
where zp is the standard normal quantile (mean 0, standard deviation 1) associated with probability p. The calculator first finds zp, then scales and shifts it using μ and σ.
How to use this calculator
Step-by-step
- Enter a probability between 0 and 1 (not including 0 or 1).
- Select whether your probability is lower-tail or upper-tail.
- Enter the distribution mean (μ).
- Enter a positive standard deviation (σ).
- Click Calculate Quantile.
You’ll get both the standard normal z-score and the final value x for your specified normal distribution.
Worked examples
Example 1: Exam percentile
Suppose test scores are normal with μ = 70 and σ = 10. What score corresponds to the 90th percentile?
- Set p = 0.90
- Choose lower-tail
- Set μ = 70, σ = 10
The result is approximately x ≈ 82.8, meaning a score around 82.8 is the 90th percentile cutoff.
Example 2: Reliability threshold
A part fails only in the worst 2% of cases. If a quality metric is normal with μ = 50 and σ = 4, and “worst” means very high values, use an upper-tail probability of 0.02.
The calculator gives the high cutoff where P(X ≥ x) = 0.02.
Common use cases
- Finding z-critical values in hypothesis testing
- Building confidence interval cutoffs
- Computing score percentiles
- Setting risk thresholds in finance
- Establishing quality control limits in manufacturing
Interpretation tips
Lower-tail vs upper-tail
These are mirror images. A lower-tail probability of 0.95 is equivalent to an upper-tail probability of 0.05 for the same x.
Check units
The returned x is in the same units as your original variable. If μ and σ are in dollars, x is dollars. If they are in points, x is points.
Frequently asked questions
Why can’t probability be exactly 0 or 1?
In a continuous normal model, exact 0 and 1 would imply negative or positive infinity for the quantile. So the inverse is only finite for probabilities strictly between 0 and 1.
Is this the same as a z-table?
Yes, conceptually. A z-table gives probabilities for z-values. The inverse normal calculator goes the other direction: from probability to z.
How accurate is this calculator?
It uses a high-quality numerical approximation (Acklam’s method), which is widely used and very accurate for practical statistics work.