Z-Score Calculator - Calculate Standard Score Instantly

A z-score, also called a standard score, is a statistical measurement that describes how far a data point lies from the mean of a distribution, measured in units of standard deviation. A z-score of 0 means the value is equal to the mean. A positive z-score indicates the value is above the mean, and a negative z-score indicates it is below the mean. The formula for the z-score is Z = (X − μ) / σ, where X is the raw data value, μ is the population mean, and σ is the population standard deviation. This simple transformation standardizes data from any distribution into a common scale, enabling direct comparisons between measurements that originally used different units or scales. Z-scores are foundational in many areas of statistics and data science. In hypothesis testing, the z-score is used as a test statistic to determine whether a sample mean differs significantly from a known population mean. In quality control, z-scores help flag measurements that fall outside acceptable ranges. In finance, they are used to assess the relative performance of stocks or portfolios, and the Altman Z-score is a well-known formula for predicting bankruptcy risk. In education, z-scores are used to standardize test results across different exams. Converting an SAT score and an ACT score to z-scores allows a direct comparison of how well two students performed relative to their respective peer groups. In healthcare, z-scores are used to track children's height and weight relative to national growth standards. Under the assumption of a normal distribution, z-scores have well-defined probability interpretations. Approximately 68% of values fall within one standard deviation of the mean (z between −1 and 1), 95% within two standard deviations, and 99.7% within three standard deviations. These percentages underpin the empirical rule widely used in statistics. When the population standard deviation is unknown, the sample standard deviation s is used instead. The resulting statistic is technically a t-score rather than a z-score, and the t-distribution should be used for inference. The z-distribution is appropriate when the standard deviation is known — which is common in quality control, standardized testing, and other fields where large historical datasets establish reliable population parameters. The calculator on this page uses the classical population formula Z = (X − μ) / σ. Enter any real number for X and μ, and any positive number for σ, to instantly obtain the z-score and a plain-English interpretation.