Carl Friedrich Gauss was a child prodigy and a brilliant mathematician who lived in the early 1800s. 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, among others.

Modern-day terminology defines the normal distribution as the bell curve with mean and variance parameters. This article explains the bell curve and applies it to trading.

### Measuring Center: Mean, Median, and Mode

Distributions can be characterized by their mean, median, and mode. The mean 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. Each of these three numbers measures the center of a distribution. For the normal distribution, however, the mean is the preferred measurement.

### Measuring Dispersion: Standard Deviation and Variance

If the values follow a normal (Gaussian) distribution, 68 percent of all scores fall within -1 and +1 standard deviations (of the mean), 95 percent fall within two standard deviations, and 99.7 percent fall within three standard deviations.

Standard deviation is the square root of the variance, which measures the spread of a distribution. (For more information on statistical analysis, read *Understanding Volatility Measures*.)

### Applying the 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.

The Gaussian distribution marked the beginning of an understanding of market probabilities. It later led to time series, Garch Models, and more applications of skew such as the Volatility Smile.

### 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. (For related reading, see *Stock Market Risk: Wagging the Tails*.)

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 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. Skewness coefficients are used to measure option prices by measuring implied volatility.