# Z-Score

## Basic Points

- Quants will use z-scores to measure how many units of standard deviation (the expected movement with 68.3% confidence) that a single instance is from the mean.
- This is generally referred to in measurements of lowercase sigma (σ). This symbol will look familiar to advanced traders because it is how volatility is expressed in formulas, such as in the Black-Scholes [theoretical options pricing] model. This σ means the same based on standard deviation whether implied (based on options prices) or realized (based on historical data).

## Expert: Quantitative Use Cases

One example of how quants use z-score is for a pair trade, which takes two separate securities and makes bets in opposite directions on them. One way that they might go about this is to measure how many units of standard deviation two normally-correlating securities are deviating. If a divergence above a z-score is greater than 2 or 3 standard deviation, then this sets the trader up for a likely convergence bet (long one short the other). However, this would only be for statistical edge (competitive trading advantage) and would be far from a guarantee. Pair trades can be dangerous and it is better to err on the side of caution by using too little than too much size.

While the probability density (which is the amount of room left under 100% chance) shrinks exponentially with each additional standard deviation, each standard deviation itself is a fixed length. And so if the standard deviation is a 2% move, and a z-score was 1.5, then that means there was a 3% move.