Probabilidade Normal (distribuição amostral)

This calculator determines the probability that the sample mean (X̄) is less than or equal to a specific value x, assuming the population follows a normal distribution with mean μ and standard deviation σ. The calculation uses the standard error of the mean (SE = σ/√n) to standardize the difference between x and μ, resulting in a z-score. The probability is then obtained from the standard normal cumulative distribution function Φ(z).

Usage is straightforward: enter the population mean (μ), population standard deviation (σ), sample size (n), and the limit value x. The calculator computes the standard error (σ/√n) and the z-score = (x − μ)/SE. The probability P(X̄ ≤ x) is then Φ(z), provided as a value between 0 and 1 (or as a percentage).

Use this tool in situations such as: estimating the chance that the average spending of a customer sample falls below a certain amount; evaluating the probability that the average crop yield is below a threshold; or checking whether a production process meets specifications based on the sample mean. It is useful in quality control, market research, and academic studies.

Cautions: the population must be normally distributed or the sample large enough (n ≥ 30) for the Central Limit Theorem to apply. The population standard deviation σ must be known; if you only have the sample standard deviation, use the Student's t-distribution. Also ensure that the value x is consistent with the mean and scale of the data.