zStatistik

zStatistik, often referred to as the z-statistic, is a standardized statistic used to measure how many standard deviations an observed value is from a specified reference value. It is based on the standard normal distribution and is widely employed in hypothesis testing, confidence interval construction, and cross-study comparison. A common form is z = (X − μ0) / σ for a single observation X with a known population mean μ0 and population standard deviation σ. For a sample mean X̄ from n observations, the standardized form is z = (X̄ − μ0) / (σ/√n) when σ is known. When σ is unknown and the sample size is small, the t-statistic is used instead of z.

Interpretation of z-statistics relies on the standard normal model. Under the null hypothesis and appropriate conditions, z follows approximately a standard normal distribution N(0,1). This allows the computation of p-values as two-sided measures 2*(1 − Φ(|z|)) and the use of fixed critical values, such as ±1.96 for a 5% significance level or ±2.58 for a 1% level.

Applications of zStatistik span several statistical practices. In hypothesis testing, it enables quick assessment of deviations from a reference value. In confidence interval estimation, z-values determine the width of intervals when σ is known. It is also used to standardize results across different studies or measurement scales, supporting comparability in meta-analyses and quality-control processes.

Limitations include the reliance on independence and, usually, a known σ or large sample size to justify the normal approximation. Outliers and non-normal data can distort z-scores, and in cases of unknown variance with small samples, the t-statistic is preferred. The concept remains a foundational tool in many statistical workflows.