Z-Scores vs Prediction Intervals

Z-Score and Prediction Intervals serve different purposes. Z-scores assess existing values within a dataset, while prediction intervals estimate the likely range for future observations.

Use Z-scores to evaluate existing values or standardize. Use prediction intervals to express uncertainty about where a new observation is likely to fall.

Comparison Table:

Feature	Z-Score	Prediction Interval
Purpose	Assess deviation from the mean	Forecast future values
Formula	$Z=\frac{x-\bar{x}}{s}$	$\bar{x}\pm t_{\alpha /2,n-1}\cdot s\cdot \sqrt{1+\frac{1}{n}}$
Distribution	Standard Normal (Z)	Student’s t-distribution
Use case	Outlier detection, normalization	Prediction of new measurements
Width of range	Based on fixed $\sigma$	Wider—accounts for both sampling error and variability
Needs population $\sigma$?	Yes (or large $n$ to approximate)	No (uses sample $s$ and $t$ for small $n$)