# Z-Score: Meaning and Formula

![Z-Score](https://www.investopedia.com/thmb/4Eba0h-H-ERNmdw8b0bbrMzUZgg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Term-Definitions_zscore.aspsketch-3c68631c0479475f9bee1d6ecb0f4e23.jpg)

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

### 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.

## 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](https://www.investopedia.com/terms/a/altman.asp) is used to calculate the likelihood that a business will go bankrupt within the range of one to 10 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.

> **Fast Fact**
>
> Z-score is also known as the standard score.

## 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](https://www.investopedia.com/terms/m/mean.asp) and [standard deviation](https://www.investopedia.com/terms/s/standarddeviation.asp) 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 standard 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 evaluation methods, such as weighting each stock's return to use [factor investing](https://www.investopedia.com/terms/f/factor-investing.asp), 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](https://www.investopedia.com/terms/v/variability.asp) (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 Calculated?

The Z-score is calculated by finding the difference between a data point and the average of the dataset, then dividing that difference by the standard deviation to see how many standard deviations the data point is from the mean.

## 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.

> **Article Sources**
>
> 1.  PennState, Eberly College of Science. "[STAT 500: 3.3.3 - Probabilities for Normal Random Variables (Z-scores)](https://online.stat.psu.edu/stat500/lesson/3/3.3/3.3.3)."
>     
> 2.  Zhao, Tianyang. "[The Relationship Between Z-Score and Stock Prices](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2595600)." April 2015, pp. 1-36. Download PDF.
>     
> 3.  Business Compass, LLC. "[Corporate Distress & Bankruptcy: Altman Z-Score+](https://businesscompassllc.com/corporate-distress-bankruptcy/)."
>     
> 4.  PennState, Eberly College of Science. "[STAT 500: 3.3.4 - The Empirical Rule](https://online.stat.psu.edu/stat500/lesson/3/3.3/3.3.4)."