Z-Score: Meaning and Formula (2024)

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 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.

ABC
1Factor (x)Mean (μ)St. Dev. (σ)
2312.176.4
31312.176.4
4812.176.4
52112.176.4
61712.176.4
71112.176.4

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
ABCD
1Factor (x)Mean (μ)St. Dev. (σ)Z-Score
2312.176.4-1.43
31312.176.40.13
4812.176.4-0.65
52112.176.41.38
61712.176.40.75
71112.176.4-0.18

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 ofvariability(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.

What Is Z-Score?

The Z-score is a way to figure out how far away a piece of data is from the average of a group, measured in standard deviations. It tells us if a data point is typical or unusual compared to the rest of the group, which is useful for spotting unusual values and comparing data between different groups.

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.

Z-Score: Meaning and Formula (2024)

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