# Trading With Gaussian Statistical Models

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.

### Key Takeaways

• 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.