The selection of a statistical score, either t or z, hinges on the knowledge of the population standard deviation. When the population standard deviation is unknown and must be estimated from the sample data, t-scores are employed. Z-scores, on the other hand, are appropriate when the population standard deviation is known. For instance, if analyzing the IQ scores of a large, well-documented population where the standard deviation is established, a z-score might be used. However, if assessing the performance of a small group of students on a new exam where the population standard deviation is unavailable, a t-score becomes more suitable.
The importance of using the correct score lies in the accuracy of statistical inferences. T-scores, compared to z-scores, account for the increased uncertainty that arises from estimating the population standard deviation. This adjustment ensures that hypothesis testing and confidence interval construction are more conservative, reducing the risk of Type I errors (false positives). Historically, the development of the t-distribution by William Sealy Gosset (under the pseudonym “Student”) addressed the limitations of using z-scores with small sample sizes and unknown population standard deviations, thereby providing a more reliable method for statistical analysis.
