How to Calculate Z-Score and Its Meaning

Z-Score

Investopedia / Tara Anand

What Is Z-Score?

Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

In investing and trading, Z-scores are measures of an instrument's variability and can be used by traders to help determine volatility. The Z-score is sometimes confused with the Altman Z-score, which is calculated using factors taken from a company's financial reports. The Altman Z-score is used to calculate the likelihood that a business will go bankrupt in the next two years, while the Z-score can be used to determine how far a stock's return differs from it's average return—and much more.

Key Takeaways

  • A Z-Score is a statistical measurement of a score's relationship to the mean in a group of scores.
  • A Z-score can reveal to a trader if a value is typical for a specified data set or if it is atypical.
  • In general, a Z-score of -3.0 to 3.0 suggests that a stock is trading within three standard deviations of its mean.
  • Traders have developed many methods that use z-score to identify correlations between trades, trading positions, and evaluate trading strategies.

Z-Score Formula

The statistical formula for a value's z-score is calculated using the following formula:

z = ( x - μ ) / σ

Where:

  • z = Z-score
  • x = the value being evaluated
  • μ = the mean
  • σ = the standard deviation

How to Calculate Z-Score

Z-Score

Calculating a z-score requires that you first determine the mean and standard deviation of your data. Once you have these figures, you can calculate your z-score. So, assume you have the following variables:

  • x = 57
  • μ = 52
  • σ = 4

You would use the variables in the formula:

  • z = ( 57 - 52 ) / 4
  • z = 1.25

So, your selected value has a z-score that indicates it is 1.25 standard deviations from the mean.

Spreadsheets

To determine z-score using a spreadsheet, you'll need to input your values and determine the average for the range and the stadard deviation. Using the formulas:

=AVERAGE(A2:A7)
=STDEV(A2:A7)

You'll find that the following values have a mean of 12.17 and a standard deviation of 6.4.

A B C
1 Factor (x) Mean (μ) St. Dev. (σ)
2 3 12.17 6.4
3 13  12.17 6.4
4 12.17 6.4
5 21  12.17 6.4
6 17 12.17 6.4
7 11 12.17 6.4

Using the z-score formula, you can figure out each factor's z-score. Use the following formula in D2, then D3, and so on:

Cell D2 = ( A2 - B2 ) / C2
Cell D3 = ( A3 - B3 ) / C3
A B C D
1 Factor (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 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

How the Z-Score Is Used

In it's most basic form, the z-score allows you determine how far (measured in standard deviations) the returns for the stock you're evaluating are from the mean of a sample of stocks. The average score you have could be the mean of a stock's annual return, the average return of the index it is listed on, or the average return of a selection of stocks you've picked.

Some traders use the z-scores in more advanced evalulation methods, such as weighting each stock's return to use factor investing, where stocks are evaluated based on specific attributes using z-scores and standard deviation. In the forex markets, traders use z-scores and confidence limits to test the capability of a trading system to generate winning and losing streaks.

Z-Scores vs. Standard Deviation

In most large data sets (assuming a normal distribution of data), 99.7% of values lie between -3 and 3 standard deviations, 95% between -2 and 2 standard deviations, and 68% between -1 and 1 standard deviations.

Standard deviation indicates the amount of variability (or dispersion) within a given data set. For instance, if a sample of normally distributed data had a standard deviation of 3.1, and another had one of 6.3, the model with a standard deviation (SD) of 6.3 is more dispersed and would graph with a lower peak than the sample with an SD of 3.1.

A distribution curve has negative and positive sides, so there are positive and negative standard deviations and z-scores. However, this has no relevance to the value itself other than indicating which side of the mean it is on. A negative value means it is on the left of the mean, and a positive value indicates it is on the right.

The z-score shows the number of standard deviations a given data point lies from the mean. So, standard deviation must be calculated first because the z-score uses it to communicate a data point's variability.

How Is Z-Score Used in Real Life?

A z-score is used in many real-life applications, such as medical evaluations, test scoring, business decision-making, and investing and trading opportunity measurements. Traders that use statistical measures like z-scores to evaluate trading opportunities are called quant traders (quantitative traders).

What Is a Good Z-Score?

The higher (or lower) a z-score is, the further away from the mean the point is. This isn't necessarily good or bad; it merely shows where the data lies in a normally distributed sample. This means it comes down to preference when evaluating an investment or opportunity. For example, some investors use a z-score range of -3.0 to 3.0 because 99.7% of normally distributed data falls in this range, while others might use -1.5 to 1.5 because they prefer scores closer to the mean.

Why Is Z-Score So Important?

A z-score is important because it tells where your data lies in the data distribution. For example, if a z-score is 1.5, it is 1.5 standard deviations away from the mean. Because 68% of your data lies within one standard deviation (if it is normally distributed), 1.5 might be considered too far from average for your comfort.

The Bottom Line

A z-score is a statistical measurement that tells you how far away from the mean (or average) your datum lies in a normally distributed sample. At its most basic level, investors and traders use quantitative analysis methods such as a z-score to determine how a stock performs compared to other stocks or its own historical performance. In more advanced z-score uses, traders weigh investments based on desirable criteria, develop other indicators, or even try to predict the outcome of a trading strategy.