Calculadora de Distribuição Normal
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
P(X ≤ x) = Φ((x − μ)/σ)
About this calculator
The Normal Distribution Calculator computes the probability P(X ≤ x) for a random variable X following a normal distribution with mean μ and standard deviation σ. It uses the error function (erf) to obtain the cumulative probability. This tool is essential for statistical analysis, quality control, and data science.
The formula used is P(X ≤ x) = Φ((x − μ)/σ), where Φ is the cumulative distribution function of the standard normal. Internally, the calculator converts x to the corresponding z-score and applies the error function to get the probability. The result is a number between 0 and 1, representing the cumulative probability.
Use this calculator when you need to determine the probability of an event in a normal distribution, such as test scores, population heights, or measurement errors. It is useful for calculating percentiles, specification limits, and critical values in hypothesis testing.
Cautions: ensure the data actually follows a normal distribution. Extreme values or small samples may invalidate results. The calculator assumes a continuous and symmetric distribution. Verify that the mean and standard deviation are correct, as errors in these parameters directly affect the probability.
Frequently asked questions
What is the difference between normal and standard normal distribution?
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be converted to standard using the z-score.
How do I interpret the result P(X ≤ x)?
P(X ≤ x) is the probability that the random variable is less than or equal to x. For example, 0.95 means a 95% chance the value is below x.
Can I use it to calculate probabilities between two values?
Yes, compute P(X ≤ b) and P(X ≤ a) and subtract: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a).
Does the calculator work for any mean and standard deviation?
Yes, as long as the standard deviation is positive. Negative or zero values are not allowed.
What does 'erf' mean in the formula?
Erf is the error function, a mathematical function used to compute the cumulative probability of the standard normal.