## Z-Score and Standard Deviation: An Overview

Although the finance industry can be complex, an understanding of the calculation and interpretation of fundamental mathematical building blocks is still the foundation for success, whether in accounting, economics, or investing.

Standard deviation and the Z-score are two such fundamentals. Z-scores can help traders gauge the volatility of securities. The score shows how far away from the mean—either above or below—a value is situated. Standard deviation is a statistical measure that shows how elements are dispersed around the average, or mean. Standard deviation helps to indicate how a particular investment will perform, so, it is a predictive calculation.

A firm grasp of how to calculate and utilize these two measurements enables a more thorough analysis of patterns and changes in any data set, from business expenditures to stock prices.

### Key Takeaways

- Standard deviation defines the line along which a particular data point lies.
- Z-score indicates how much a given value differs from the standard deviation.
- The Z-score, or standard score, is the number of standard deviations a given data point lies above or below mean.
- Standard deviation is essentially a reflection of the amount of variability within a given data set.
- Bollinger Bands are a technical indicator used by traders and analysts to assess market volatility based on standard deviation.

## Z-Score

The Z-score, or standard score, is the number of standard deviations a given data point lies above or below the mean. The mean is the average of all values in a group, added together, and then divided by the total number of items in the group.

To calculate the Z-score, subtract the mean from each of the individual data points and divide the result by the standard deviation. Results of zero show the point and the mean equal. A result of one indicates the point is one standard deviation above the mean and when data points are below the mean, the Z-score is negative.

In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above or below the mean.

Z-scores offer analysts a way to compare data against a norm. A given company’s financial information is more meaningful when you know how it compares to that of other, comparable companies. Z-score results of zero indicate that the data point being analyzed is exactly average, situated among the norm. A score of 1 indicates that the data are one standard deviation from the mean, while a Z-score of -1 places the data one standard deviation below the mean. The higher the Z-score, the further from the norm the data can be considered to be.

In investing, when the Z-score is higher it indicates that the expected returns will be volatile, or are likely to be different from what is expected.

A Bollinger Band® is a technical indicator used by traders and analysts to assess market volatility based on standard deviation. Simply put, they are a visual representation of the Z-score. For any given price, the number of standard deviations from the mean is reflected by the number of Bollinger Bands between the price and the exponential moving average (EMA).

## Standard Deviation

Standard deviation is essentially a reflection of the amount of variability within a given data set. It shows the extent to which the individual data points in a data set vary from the mean. In investing, a large standard deviation means that more of your data points deviate from the norm, so the investment will either outperform or underperform similar securities. A small standard deviation means that more of your data points are clustered near the norm and returns will be closer to the expected results.

Investors expect a benchmark index fund to have a low standard deviation. However, with growth funds, the deviation should be higher as the management will make aggressive moves to capture returns. As with other investments, higher returns equate to higher investment risks.

The standard deviation can be visualized as a bell curve, with a flatter, more spread-out bell curve representing a large standard deviation and a steep, tall bell curve representing a small standard deviation.

To calculate the standard deviation, first, calculate the difference between each data point and the mean. The differences are then squared, summed, and averaged to produce the variance. The standard deviation, then, is the square root of the variance, which brings it back to the original unit of measure.

## The Bottom Line

In investing, standard deviation and the Z-score can be useful tools in determining market volatility. As the standard deviation increases, it indicates that price action varies widely within the established time frame. Given this information, the Z-score of a particular price indicates how typical or atypical this movement is based on previous performance.