Z-⁠Score

What is a z-score?

A z-score is a standardized value that expresses an observation’s signed distance from a reference mean in standard-deviation units. Positive values sit above the mean, negative values below it and zero at the mean. Because every series shares that scale, traders use z-scores to compare unlike prices, returns or spreads and to flag unusually stretched readings.

Compute z = (x − mu) / sigma with a positive reference standard deviation. A zero divisor makes the score undefined.

How z-scores map to percentiles

Standardization and percentile calculation are separate operations. A standardized variable with finite positive variance has mean zero and variance one; a normal variable becomes standard normal.

On a normal reference curve, +2.5 is the 99.38th percentile. The upper-tail probability is 0.62%. At +2 the upper tail is 2.28%.

At +4, the normal upper tail holds 3.2 of every 100,000 outcomes, and the two tails outside ±4 hold 6.3 per 100,000. State which tail the calculation uses.

Scores from different distributions share a distance scale. Their percentiles come from those distributions or their empirical ranks. Estimating a mean and standard deviation from a sample does not by itself establish a standard normal predictive distribution.

Reference windows and composite z-score screens

Choose prices, returns, spreads or another metric and keep that choice consistent. Specify the window and standard-deviation divisor in the calculation. Using prior observations provides a reference available before the current observation.

For a composite screen, standardize each input and set its weight. Correlation between inputs affects the composite’s scale, so an average of z-scores is not automatically a standard normal variable.

How traders use z-scores for mean reversion

A mean-reversion rule can use distance thresholds for entry and a specified exit toward its reference. A trend rule can use the same scores to rank strength. Direction and execution come from the strategy definition.

Limitations of the z-score

• Convert a z-score to a probability only through a distribution that fits the data. The score gives distance; normal percentiles apply only when the data fits a normal curve, and fat-tailed series reach large scores more often than those percentiles imply.
• State the series, window and standard-deviation divisor with every score, because the same price can carry different z-scores under different choices.
• Check the trend before you fade a stretched reading. The reference mean and standard deviation are estimates that change over time, and in a trend a stretched reading can stay stretched instead of reverting.
• Set a minimum standard deviation for the screen. A very small one makes the score swing sharply, and a zero one leaves it undefined.
• Calibrate composite thresholds on their own. An average of correlated z-scores is not automatically a standard normal variable.

Bottom line

A z-score turns any observation into a distance from its mean in standard deviations, which makes stretched readings easy to spot and compare. State the series and window, convert scores to probabilities only through a distribution that fits the data, and let the strategy rules decide direction and exits.