What statistical value is determined by the difference between the observed values and their mean?

The correct answer is the residual. In statistics, a residual represents the difference between the observed value and the predicted value (often the mean, in a simple case) of a dataset. It is calculated by subtracting the predicted value from the observed value. This concept is critical in regression analysis and helps in assessing the accuracy of a model.

Residuals provide insight into the nature of the errors in a model. A smaller residual indicates a better fit between the model and the observed data, while larger residuals suggest the model may not accurately represent the data. Understanding residuals is essential for refining statistical models and improving predictions.

The other options, while related to statistical measures, refer to different concepts. For example, standard deviation quantifies the amount of variation or dispersion in a set of values, rather than the difference between an individual observation and the mean. Error margin pertains to the uncertainty in an estimation, often used in the context of confidence intervals, and does not specifically measure the residuals. Data variability generally refers to how spread out the data points are in a dataset but does not focus on the individual differences between observed values and their means.