InvNorm Calculator
Convert percentiles into z-scores and x-values, find middle-area ranges, and get confidence-level critical values for any normal distribution.
Example: 95 means find x where P(X ≤ x) = 0.95.
What InvNorm Means
InvNorm stands for inverse normal distribution. Instead of starting with a z-score or x-value and finding the area under the curve, you start with a probability and work backward to the cutoff value. This is useful when a statistics class gives you a percentile, a confidence level, or a target tail probability and asks for the matching z-score or data value.
Percentiles
Find the score that cuts off the bottom 80th, 90th, 95th, or any other percentile in a normal distribution.
Middle Areas
Find the symmetric interval around the mean that contains a chosen percentage of all values, such as the central 95%.
Confidence Levels
Get the critical z* values used in confidence intervals, margin-of-error formulas, and many introductory statistics problems.
Key Formulas Behind the Calculator
Standard normal inverse
z = Φ⁻¹(p)
This converts a cumulative probability p into a standard-normal z-score.
Convert z to any normal distribution
x = μ + zσ
Once you know z, shift by the mean and scale by the standard deviation.
Two-sided confidence level
z* = Φ⁻¹(1 - α / 2)
Here α = 1 - confidence level. For 95%, α = 0.05 and z* ≈ 1.96.
Central interval
tail = (1 - c) / 2
The lower bound uses Φ⁻¹(tail) and the upper bound uses Φ⁻¹(1 - tail).
Common InvNorm Examples
| Problem | Input | Result |
|---|---|---|
| 90th percentile of Z | p = 0.90 | z ≈ 1.2816 |
| 95% confidence level | CL = 0.95 | z* ≈ 1.9600 |
| 95th percentile of N(100, 15) | p = 0.95 | x ≈ 124.67 |
Frequently Asked Questions
What does an invnorm calculator do?
An invnorm calculator finds the value x or z-score that corresponds to a given cumulative probability in a normal distribution. For example, the 95th percentile of the standard normal distribution is about z = 1.645 because 95% of values lie below it.
How do you find a z-score from a percentile?
Convert the percentile to a decimal probability between 0 and 1, then apply the inverse standard normal function z = Φ⁻¹(p). For example, the 90th percentile means p = 0.90, so z ≈ 1.2816.
What is the difference between invnorm and normalcdf?
normalcdf starts with x-values and returns a probability or area. invnorm does the reverse: it starts with a probability and returns the corresponding x-value or z-score. They are inverse operations for the normal distribution.
What z-score corresponds to a 95% confidence level?
For a two-sided 95% confidence level, the critical value is z* ≈ 1.96. For a one-sided 95% level, the critical value is about 1.645. Our calculator uses the two-sided convention for confidence-level mode.
Can I use invnorm with a non-standard normal distribution?
Yes. If your distribution has mean μ and standard deviation σ, first find the standard normal z-score from the probability, then convert it with x = μ + zσ. That is why the calculator accepts custom mean and standard deviation inputs.
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