ZTväärtus

ZTväärtus (Z-value) is a standardized measure that indicates how many standard deviations a value X is from the mean of a distribution, typically the standard normal distribution N(0,1). It provides a way to compare observations from different distributions on a common scale.

Calculation can be expressed in several forms. For a single observation X drawn from a population with mean μ and standard deviation σ, the ZTväärtus is (X − μ) / σ. For a sample mean X̄ used in a test with known σ, the z-statistic is (X̄ − μ0) / (σ / √n). If the population standard deviation is unknown, the ZTväärtus is generally replaced by a t-statistic, which uses the sample standard deviation and follows a t-distribution with n−1 degrees of freedom.

ZTväärtus follows the standard normal distribution N(0,1). This allows converting observations into p-values and comparing results across different samples. A ZTväärtus near 0 indicates the observation is close to the mean; larger absolute values indicate values that are rarer under the assumed normal model.

Common uses include hypothesis testing with Z-tests, constructing confidence intervals, and identifying critical values at chosen significance levels. For example, in a two-tailed test at the 0.05 level, critical absolute ZTväärtus values are about 1.96; at the 0.01 level, about 2.58.

Limitations include the assumption of normality or a large-sample approximation. When these conditions are not met, the ZTväärtus may be inappropriate, and alternative methods such as the t-statistic or nonparametric approaches should be considered.