Z-Score: Meaning and Formula

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. In investing and trading, Z-scores are measures of an instrument's variability and can be used by traders to help determine volatility.

Understanding Z-Score

Z-score is a statistical measure that quantifies the distance between a data point and the mean of a dataset. It's expressed in terms of standard deviations. It indicates how many standard deviations a data point is from the mean of the distribution.

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.

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.

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.

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

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.

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.