Z-Score

A z-score, also called a standard score, states how far a single value is from the average of its group, measured in units of standard deviation. A z-score of zero means the value equals the mean. A score of 2 means it is two standard deviations above, and a score of -1.5 means it is one and a half below. Converting raw numbers to z-scores is called standardization.

The measure allows values from different scales to be compared directly and shows at a glance how unusual an observation is. In finance it is used to judge whether a price, spread or return is extreme relative to its history, to put different investment signals on a common footing and to flag outliers in data. The name comes from the letter conventionally used for a standard normal variable.

The calculation is:

• $x$ is the value being assessed.
• $\mu$ is the mean of the population or reference sample.
• $\sigma$ is the standard deviation of that population or sample.

Suppose a share's daily returns have averaged 0.05 percent with a standard deviation of 1.30 percent, and today it falls 3.20 percent. The z-score is $(-3.20 - 0.05) / 1.30 = -3.25 / 1.30 = -2.5$. The fall is two and a half standard deviations below the average day.

If the data follow a normal distribution, z-scores follow the standard normal distribution and translate directly into probabilities. About 68 percent of values have a score between -1 and 1, about 95 percent between -2 and 2, and about 99.7 percent between -3 and 3. A score beyond 2 in either direction would occur about 4.6 percent of the time, and beyond 3 about 0.27 percent.

Those probabilities hold only under normality. Market returns have fat tails, so large z-scores occur far more often than the table suggests. A move of -2.5 is unusual but not rare for a share, and scores of 5 or more, which the normal curve treats as practically impossible, are seen in most markets every few years. Such an outcome is sometimes called a six sigma event.

• Mean-reversion signals. In pairs trading and statistical arbitrage, the spread between two related instruments is converted to a z-score using its mean and standard deviation over a rolling window. A common rule opens a position when the score passes 2 in either direction and closes it when the score returns toward zero, on the expectation of mean reversion.
• Chart indicators. Bollinger Bands plot lines two standard deviations above and below a moving average. A price touching the upper band has a z-score of 2 against that average.
• Combining factors. In factor investing, measures such as valuation, profitability and past return are in different units. Each is converted to a z-score across the universe of shares on a given date, and the scores are added to form a composite ranking. Extreme scores are usually capped at 3 so that a single outlier does not dominate.
• Data checks and surveillance. Observations with very large scores are flagged for review as possible errors or as unusual activity.

For a time series, the mean and standard deviation are normally calculated over a moving window, such as the past 60 days, and the choice of window changes the result. A short window adapts quickly but makes almost any sustained move look normal after a few days. A long window is slow to recognise a lasting change in level.

The statistics must be calculated only from data available at the time. Standardizing with the mean and standard deviation of the whole history, including later observations, introduces look-ahead bias into a backtest. Because the mean and standard deviation are themselves distorted by extreme values, a robust version replaces them with the median and the median absolute deviation.

A z-score assumes there is a stable average to measure against. If the underlying series lacks stationarity, for instance because the relationship between two shares has changed permanently, a large score signals a new regime and not a temporary deviation, and a mean-reversion trade based on it will lose. The score also says nothing about why a value is extreme. A spread may be wide because of a temporary imbalance or because of news that justifies it.

The name is shared by an unrelated measure. The Altman Z-score, published by Edward Altman in 1968, is a formula that combines five financial ratios from a company's accounts into a single number used to assess the likelihood of bankruptcy. For the original model of listed manufacturing companies, a score above about 3 indicates financial strength and a score below about 1.8 indicates distress. It is a credit analysis tool and not a standardized score in the statistical sense. The z-test, by contrast, is a direct application of the statistical z-score to a sample average.