Carl Friedrich Gauss was a child prodigy and a brilliant mathematician who lived from the late 18th to mid-19th century. Gauss' contributions included quadratic equations, least squares analysis, and the normal distribution. Although the normal distribution was known from the writings of Abraham de Moivre as early as the mid-1700s, Gauss is often given credit for the discovery, and the normal distribution is often referred to as the Gaussian distribution.
Much of the study of statistics originated from Gauss, and his models are applied to financial markets, prices, and probabilities. Modern-day terminology defines the normal distribution as the bell curve, with mean and variance parameters. This article explains the bell curve and applies the concept to trading.
Measuring Center: Mean, Median, and Mode
Measures of the center of a distribution include the mean, median, and mode. The mean, which is simply an average, is obtained by adding all scores and dividing by the number of scores. The median is obtained by adding the two middle numbers of an ordered sample and dividing by two (in case of an even number of data values), or simply just taking the middle value (in case of an odd number of data values). The mode is the most frequent of the numbers in a distribution of values.
- Gaussian distribution is a statistical concept that is also known as the normal distribution.
- For a given set of data, the normal distribution puts the mean (or average) at the center and standard deviations measure dispersion around the mean.
- In a normal distribution, 68% of all data fall between -1 and +1 standard deviations of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
- Investments with high standard deviations are considered higher risk compared to those with low standard deviations.
Theoretically, the median, mode, and mean are identical for a normal distribution. However, when using data, the mean is the preferred measurement of the center among these three. If the values follow a normal (Gaussian) distribution, 68% of all scores fall within -1 and +1 standard deviations (of the mean), 95% fall within two standard deviations, and 99.7% fall within three standard deviations. Standard deviation is the square root of the variance, which measures the spread of a distribution.
Gaussian Model to Trading
Standard deviation measures volatility and determines what performance of returns can be expected. Smaller standard deviations imply less risk for an investment while higher standard deviations imply higher risk. Traders can measure closing prices as the difference from the mean; a larger difference between the actual value and the mean suggests a higher standard deviation and, therefore, more volatility.
Prices that deviate far away from the mean might revert back to the mean, so that traders can take advantage of these situations, and prices that trade in a small range might be ready for a breakout. The often-used technical indicator for standard deviation trades is the Bollinger Band® because it is a measure of volatility set at two standard deviations for upper and lower bands with a 21-day moving average.
Skew and Kurtosis
Data do not usually follow the precise bell curve pattern of the normal distribution. Skewness and kurtosis are measures of how data deviate from this ideal pattern. Skewness measures the asymmetry of the tails of the distribution: A positive skew has data that deviate farther on the high side of the mean than on the low side; the opposite is true for negative skew.
While skewness relates to the imbalance of the tails, kurtosis is concerned with the extremity of the tails regardless of whether they are above or below the mean. A leptokurtic distribution has positive excess kurtosis and has data values that are more extreme (in either tail) than predicted by the normal distribution (e.g., five or more standard deviations from the mean). A negative excess kurtosis, referred to as platykurtosis, is characterized by a distribution with extreme value character that is less extreme than that of the normal distribution.
As an application of skewness and kurtosis, the analysis of fixed income securities, for example, requires careful statistical analysis to determine the volatility of a portfolio when interest rates vary. Models that predict the direction of movements must factor in skewness and kurtosis to forecast the performance of a bond portfolio. These statistical concepts can be further applied to determine price movements for many other financial instruments such as stocks, options, and currency pairs.