Z Score (simple)

The Z-Score measures how many standard deviations a value is from the mean. In trading, it quantifies how "unusual" the current price is relative to recent history, enabling mean-reversion strategies.

$$Z = \frac{X - \mu}{\sigma}$$

Where:

• $X$ = Current value
• $\mu$ = Mean over the lookback period
• $\sigma$ = Standard deviation over the lookback period

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How It Works

flowchart TD
    subgraph Input["Price Data"]
        P["Prices over n periods"]
    end
    
    subgraph Stats["Statistical Calculation"]
        MEAN["μ = Mean of prices"]
        STD["σ = Standard deviation"]
        CURR["X = Current price"]
    end
    
    subgraph Calc["Z-Score"]
        DIFF["X - μ"]
        DIV["÷ σ"]
        Z["Z-Score"]
    end
    
    P --> MEAN & STD
    CURR & MEAN --> DIFF
    DIFF & STD --> DIV --> Z

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Missing Data

If any value the calculation needs at a position is missing (null) — a data gap or a closed-market day — the Z-Score at that position is null. The math is never faked with substitute values: a gap in your inputs stays a gap in the output.

Separate from missing data, the Z-Score is mathematically undefined when the standard deviation is zero — a flat series has no "distance from the mean" to measure — so that position is also null, never a fabricated 0 that would falsely read as "exactly at the mean".

X	μ	σ	Z-Score
null	100	5	null
110	null	5	null
110	100	0	null
110	100	5	2.0

In live execution the module returns exactly one datapoint — the newest position — which is null when it cannot be determined.

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Interpreting Z-Score

Z-Score	Interpretation
> +2	Significantly above average (potential overbought)
+1 to +2	Above average
-1 to +1	Normal range
-2 to -1	Below average
< -2	Significantly below average (potential oversold)

flowchart LR
    subgraph Zones["Z-Score Distribution"]
        EXT2[">+2 σ<br/>Extreme high"]
        HIGH["+1 to +2 σ<br/>Above normal"]
        NORM["-1 to +1 σ<br/>Normal (68%)"]
        LOW["-2 to -1 σ<br/>Below normal"]
        EXT1["<-2 σ<br/>Extreme low"]
    end
    
    EXT2 --> HIGH --> NORM --> LOW --> EXT1

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Trading Applications

1. Mean Reversion Strategy

Z-Score	Signal
< -2	Consider buying (price abnormally low)
> +2	Consider selling (price abnormally high)
Approaching 0	Take profits (returning to mean)

2. Pairs Trading

Compare Z-Scores of two correlated assets:

• Diverging Z-Scores → Trade the spread
• Converging Z-Scores → Close position

3. Risk Management

Flag positions where Z-Score exceeds thresholds for review or automatic stop-loss.

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Statistical Context

The Z-Score assumes a normal distribution:

Z-Score Range	Data Within Range
±1σ	~68%
±2σ	~95%
±3σ	~99.7%

In practice, financial data often has "fat tails," meaning extreme events occur more frequently than a normal distribution predicts.

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Parameters

Parameter	Common Values	Description
Lookback	20, 50, 100	Periods for mean and std dev

Shorter lookbacks = more reactive but noisier
Longer lookbacks = smoother but slower to adapt

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Z-Score vs Other Normalization

Method	Formula	Use Case
Z-Score	(X - μ) / σ	Mean reversion, outlier detection
Min-Max	(X - min) / (max - min)	Bounded 0-1 scaling
Percentile	Rank / N	Distribution-free ranking

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Advantages

• Standardized — Comparable across different assets
• Statistical foundation — Rooted in probability theory
• Mean reversion — Ideal for reversion strategies
• Outlier detection — Identifies unusual price behavior

Disadvantages

• Assumes normality — Markets often have fat tails
• Backward-looking — Mean and std dev from history
• Parameter sensitive — Lookback period matters
• Trending markets — May give premature signals

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Further Reading

• Z-Score in Trading - Investopedia
• Mean Reversion Strategies