Carl Friedrich Gauss was a brilliant mathematician who lived in the early 1800s and gave the world quadratic equations, least squares analysis, and is also given credit for the normal distribution. Though the normal distribution was known from the writings of Abraham de Moivre as early as the mid-1700s, Gauss is often given the 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, allowing us to understand markets, prices and probabilities, among other applications.

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

### Measuring Center: Mean, Median and Mode

Distributions can be characterized by their mean, median and mode, among other measurements. The mean is obtained by adding all scores and dividing by the number of scores to obtain the average. 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 “center” of a distribution, but for the normal distribution, the mean is the preferred measurement.

### Measuring Dispersion: Standard Deviation and Variance

If the values follow a normal (Gaussian) distribution, we will find that 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 how spread out our distribution is. (For more information on statistical analysis, check out "Understanding Volatility Measures.")

### Applying the Gaussian Model to Trading

Standard deviation measures volatility and tells investors what kind of performance of returns can be expected. Smaller standard deviations mean less risk for an investment, while higher standard deviations mean a higher risk. Traders can measure closing prices as difference from the mean; a larger difference between the actual value and the mean suggests higher standard deviation and therefore more volatility.

Prices that deviate far away from the mean might revert back to the mean, so that traders might be able to take advantage of these situations. And prices that trade in a small range might ready for a breakout. The often-used technical indicator for standard deviation trades is the Bollinger Band®, because they are a measure of volatility set at two standard deviations for upper and lower bands with a 21-day moving average.

The Gaussian distribution was just the beginning of understanding market probabilities. It later led to Time Series and Garch Models, as well as more applications of skew such as the Volatility Smile.

### Skew and Kurtosis: How the Gaussian Model Goes Wrong

Data do not usually follow the precise bell curve pattern of the normal distribution. Skewness and kurtosis are measures of how the data deviate from this ideal pattern. Skewness measures 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 concerns the extremity of the tails in general, no matter whether 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 less extreme than that of the normal distribution.

As an application of skewness and kurtosis, analysis of fixed income securities requires careful statistical analysis to determine the volatility of a portfolio when interest rates vary. Models to 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.

Standard deviation measures volatility and asks what kind of performance returns can be expected. Smaller standard deviations may mean less risk for a stock, while higher volatility may mean a higher level of uncertainty. Traders can measure closing prices from the average as it is dispersed from the mean. Dispersion would then measure the difference from actual value to average value. A larger difference between the two means a higher standard deviation and volatility. Prices that deviate far away from the mean often revert back to the mean, so that traders can take advantage of these situations. And prices that trade in a small range are ready for a breakout.

The often-used technical indicator for standard deviation trades is the Bollinger Band®, because they are a measure of volatility set at 2 standard deviations for upper and lower bands with a 21-day moving average. The Gauss Distribution was just the beginning of understanding market probabilities. It later led to Time Series and Garch Models, as well as more applications of skew such as the Volatility Smile.