A Case for the T-statistic

The z-score offers a simple method for point-anomaly detection by measuring how many standard deviations an observation is from the mean. Its primary limitation is the assumption that the true population standard deviation is known, which is rarely the case in real-world applications. The t-statistic provides a more robust alternative, especially for small samples, as it accounts for the uncertainty introduced by estimating the standard deviation from the data. Both statistics are framed as a signal-to-noise ratio, where the t-statistic correctly uses the standard error as the noise term for estimated parameters like a regression slope. The analysis also demonstrates the critical impact of implementation choices, such as whether to include the current data point in a rolling window calculation for anomaly detection.