Calculadora de Distribuição Normal Inversa (z-crítico)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
z = Φ⁻¹(p)
About this calculator
The Inverse Normal Distribution Calculator finds the critical z-value (z-score) corresponding to a cumulative probability p in the standard normal distribution. In other words, given a probability value (between 0 and 1), it returns the point on the normal curve where the area to the left equals p. The calculation is done numerically using binary search with the error function (erf) for high precision.
This tool is essential for hypothesis testing and constructing confidence intervals. For example, for a 95% confidence level, the two-tailed critical z is approximately 1.96. It is also useful for calculating percentiles in normal distributions, such as the z-score for the 90th percentile (z ≈ 1.28).
Use the calculator when you need to determine the cutoff point in a normal distribution for a given probability. Be careful: ensure the entered probability is in decimal format (between 0 and 1). Values outside this range will generate an error. Also, remember that the standard normal distribution has mean 0 and standard deviation 1; for other means or standard deviations, standardization is required.
The calculator uses binary search to find z with a precision of 10⁻¹². The error function (erf) is implemented with a high-quality approximation, ensuring reliable results for statistical, scientific, and engineering applications.
Frequently asked questions
What is the critical z-value?
It is the value on the standard normal distribution that separates the rejection region from the non-rejection region in a hypothesis test, or defines the limits of a confidence interval.
How do I use the calculator for a 95% confidence interval?
For a two-tailed interval, enter p = 0.975 (area to the left of the upper limit) or p = 0.025 (for the lower limit). The result will be approximately 1.96 and -1.96.
Can I use the calculator for non-standard normal distributions?
Yes, but you need to standardize first: subtract the mean and divide by the standard deviation. The obtained z can be converted back.
What is the precision of the calculation?
The binary search finds z with a precision of 10⁻¹², using the error function (erf) with a high-quality approximation.
What happens if I enter a probability outside [0,1]?
The calculator will display an error message, as the probability must be between 0 and 1 (inclusive).