Although the finance industry can be complex, an understanding of the calculation and interpretation of basic mathematical building blocks is still the foundation for success, whether in accounting, economics or investing. Standard deviation and Z-score are two such fundamentals. 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.
Standard deviation is essentially a reflection of the amount of variability within a given data set. To calculate 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 is simply the square root of the variance, which brings it back to the original unit of measure.
The Z-score, by contrast, is the number of standard deviations a given data point lies from the mean. To calculate Z-score, simply subtract the mean from each data point and divide the result by the standard deviation. For data points that 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 and below the mean.
In investing, standard deviation and 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.
Bollinger Bands are 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).