Z-Score¶
A z-score measures how many standard deviations a data point sits above or below the mean of its distribution. Calculated as z = (x − μ) / σ, it converts raw prices, spreads, or indicator values into a normalized scale. In trading, z-scores give "unusual" a precise, comparable definition across instruments with different prices and volatilities.
HOW IT WORKS¶
The formula divides the difference between the current observation and the rolling mean by the rolling standard deviation over the same lookback window. Common windows are 20, 50, or 100 periods. Under a normal distribution, a z-score beyond ±2 occurs roughly 2.3% of the time on either tail; beyond ±3, less than 0.15%. Financial returns are fat-tailed, so extremes appear more often than those probabilities imply — a z of ±2 is a useful signal threshold, but a z of ±4 or ±5 is not a near-impossible event. The z-score is descriptive, not predictive: it measures distance from the mean, not the probability of reversion. Reversion is only a valid expectation when the underlying series is stationary.
Common thresholds used in practice
| Z-Score | Interpretation |
|---|---|
| 0 | At the rolling mean |
| ±1 | Mildly stretched (~84th / 16th percentile) |
| ±2 | Typical entry threshold for mean-reversion |
| ±3 | Stop-loss or "relationship broken" level |
IN TAPEBOARD¶
Tapeboard displays rolling z-scores on price series, spreads, and ratios directly in the terminal. When building a pairs trade, the spread chart overlays the z-score so entry and exit levels at ±2 and 0 are visible without manual calculation. The cross-asset scanner lets traders filter for instruments whose price z-score breaches a user-defined threshold across any lookback window. Risk monitors on the portfolio dashboard track z-scores of factor exposures, flagging when a position has stretched beyond historical norms before the move registers in P&L.