zórták

Z-érték, commonly known as the z-score, is a statistic that describes how far a value is from the mean of a dataset, measured in units of the data’s standard deviation. For a population value X it is z = (X − μ) / σ, where μ is the population mean and σ the population standard deviation. For a sample value, z = (X − x̄) / s, where x̄ is the sample mean and s the sample standard deviation.

If the underlying data are normally distributed with mean μ and standard deviation σ, the z-score follows a standard normal distribution N(0,1). This standardization enables comparison across different datasets and scales, and it underpins many statistical procedures.

Interpretation is straightforward: z = 0 indicates a value at the mean; positive z-values indicate values above the mean, negative values below. The magnitude |z| shows the distance from the mean in standard deviation units. In a standard normal distribution, about 95% of values lie within roughly ±1.96 and about 99.7% within roughly ±3.0.

Applications include hypothesis testing (z-tests when σ is known), construction of confidence intervals for means (using z), and detecting outliers or scoring across different scales. In practice, when σ is unknown and the sample size is small, the t-distribution is used instead of the z-distribution. For non-normal data, z-scores remain a useful standardization but their interpretive power is reduced, and robust methods may be preferred.

Z-érték is a common tool in statistics and data analysis for standardizing and comparing values across different contexts.