Introduction to Z-Score
Z-score is a statistical measurement that compares a value’s position relative to the mean of a group of values, expressed in terms of standard deviations from the mean. In the realms of investing and trading, Z-scores are critical for assessing an instrument’s variability and helping traders determine volatility.
Key Insights
- A Z-Score gauges a score’s relationship with the mean within a group of scores.
- It helps traders identify whether a value is typical or atypical for a given dataset.
- Generally, a Z-score between -3.0 and 3.0 implies a stock is trading within three standard deviations of its mean.
- Traders employ Z-scores to detect correlations between trades, positions, and optimize their trading strategies.
Understanding Z-Score
Z-score quantifies the distance between a data point and the mean of a dataset, expressed in standard deviations. It indicates how many standard deviations a data point is from the mean. A Z-score of 0 means the data point is identical to the mean score. Positive and negative Z-scores indicate scores above and below the mean, respectively. The Z-score, also known as the standard score, shouldn’t be confused with the Altman Z-score, which predicts corporate bankruptcy probability.
Z-Score Formula
The Z-Score is calculated using the formula:
$$ Z = rac{(x - ext{mean})}{ ext{standard deviation}} $$
Here,
- Z = Z-score
- x = the value under evaluation
- mean = average of the dataset
- standard deviation = data variability measure
How to Calculate Z-Score
Manual Calculation
To calculate a Z-score, first determine your data’s mean and standard deviation. Let’s say,
- x = 57
- mean = 52
- standard deviation = 4
Using the formula:
- Z = (57 - 52) / 4
- Z = 1.25
So, the value has a Z-score of 1.25, placing it 1.25 standard deviations from the mean.
Using Spreadsheets
In a spreadsheet, input values and use formulas to find the average and standard deviation of the dataset. Use:
- =AVERAGE(A2:A7)
- =STDEV(A2:A7)
Given the values, the mean is 12.17 and the standard deviation is 6.4.
| A | B | C | |
|---|---|---|---|
| 1 | Value (x) | Mean (μ) | St. Dev. (σ) |
| 2 | 3 | 12.17 | 6.4 |
| 3 | 13 | 12.17 | 6.4 |
| 4 | 8 | 12.17 | 6.4 |
| 5 | 21 | 12.17 | 6.4 |
| 6 | 17 | 12.17 | 6.4 |
| 7 | 11 | 12.17 | 6.4 |
Enter the Z-score calculation formula in D2, D3, and so forth:
- D2 = ( A2 - B2 ) / C2
| A | B | C | D | |
|---|---|---|---|---|
| 1 | Value (x) | Mean (μ) | St. Dev. (σ) | Z-Score |
| 2 | 3 | 12.17 | 6.4 | -1.43 |
| 3 | 13 | 12.17 | 6.4 | 0.13 |
| 4 | 8 | 12.17 | 6.4 | -0.65 |
| 5 | 21 | 12.17 | 6.4 | 1.38 |
| 6 | 17 | 12.17 | 6.4 | 0.75 |
| 7 | 11 | 12.17 | 6.4 | -0.18 |
Applications of Z-Score
Basic application of Z-scores allows you to determine the standard deviations between stock returns and a mean of sample stocks—whether it’s the mean annual return, index mean, or mean return of a stock selection. Traders may also use Z-scores in more advanced evaluations, like factor investing, or to test a trading system’s potential for generating streaks through confidence limits.
Z-Scores vs. Standard Deviation
In normally distributed large datasets, 99.7% of values lie between -3 and 3 standard deviations, 95% between -2 and 2, and 68% between -1 and 1 standard deviations. Standard deviation (SD) measures data dispersion within a set, illustrating variability.
A Z-score simply indicates the standard deviations a specific data point is from the mean, highlighting its relevance, but requiring initial standard deviation calculation.
Analyzing Z-Score in Real Life
Medical Evaluations
In healthcare, Z-scores can evaluate patient metrics against population norms to detect anomalous data.
Test Scoring
Education systems utilize Z-scores to compare individual student performances to overall test performances.
Business Decisions
Z-scores enable businesses to assess financial trends, enhancing decision-making processes.
Investment Strategies
Quant traders harness statistical measures like Z-scores to pinpoint opportune trades.
Determining a ‘Good’ Z-Score
Whether high or low, Z-scores signify differing degrees from the mean, which is subjective based on analysis needs. For investment evaluation, preference varies; some prefer closer to the mean, while others are comfortable with wider variability.
Importance of Z-Score
The Z-score indicates data distribution positioning, essential for analyses like stock performance comparison and identifying trends against historical performance or among competitive stocks.
Final Thoughts
Z-scores, integral to statistical analysis, inform investors and traders on how unlike stocks or returns compare within a sample or historical data. Their versatility extends into sophisticated evaluations, influencing weighted investments, additional indicators creation, or trading strategy forecasts.
Related Terms: Standard Deviation, Mean, Altman Z-score, Factor Investing, Quantitative Trading.
