Carl Friedrich Gauss was a child prodigy and a talented mathematician who lived from the end of the 18th century to the middle of the 19th century. Gauss’ contributions include quadratic equations, least squares analysis, and normal distribution. Although the normal distribution was known in the works of Abraham de Moivre as early as the mid-1700s, Gaussian is often credited with this discovery, and the normal distribution is often referred to as the Gaussian distribution.

Many statistical studies originated from Gauss, and his models were applied to financial markets, prices, and probabilities. Modern terminology defines the normal distribution as a bell-shaped curve with mean and variance parameters. This article explains the bell curve and applies the concept to trading.

## Measurement center: mean, median and mode

The measurement of the center of the distribution includes the mean, median, and mode. The average value is just an average value, which is obtained by adding up all the scores and dividing by the number of scores. The median is obtained by adding two intermediate numbers of an ordered sample and dividing by two (in the case of an even number of data values), or taking only the intermediate value (in the case of an odd number of data values). The mode is the most common number in the value distribution.

Key points

- The Gaussian distribution is a statistical concept, also known as the normal distribution.
- For a given set of data, the normal distribution places the mean (or mean) in the center, and the standard deviation measures the dispersion around the mean.
- In a normal distribution, 68% of the data fall within -1 to +1 standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
- Compared with investments with low standard deviations, investments with high standard deviations are considered riskier.

Theoretically, the median, mode, and mean of a normal distribution are the same. However, when using data, the average is the preferred metric for these three centers. If the values follow a normal (Gaussian) distribution, 68% of all scores fall within -1 and +1 standard deviations (mean), 95% fall within two standard deviations, and 99.7% fall within three standard deviations . The standard deviation is the square root of the variance, which measures the spread of the distribution.

## Gaussian trading model

The standard deviation measures volatility and determines the expected return performance. A smaller standard deviation means less investment risk, and a higher standard deviation means higher risk. Traders can measure the closing price by the difference from the mean; a large difference between the actual value and the average indicates a higher standard deviation and therefore greater volatility.

Prices far away from the mean may return to the mean so that traders can take advantage of these situations, while prices traded within a small range may be ready for a breakout. The commonly used technical indicator for standard deviation trading is Bollinger Band®, because it is a measure of volatility, which is set to have two standard deviations with the upper and lower limits of the 21-day moving average.

## Skewness and kurtosis

The data usually does not follow the exact bell curve pattern of a normal distribution. Skewness and kurtosis are indicators of how data deviates from this ideal pattern. Skewness measures the asymmetry of the tail of the distribution: data with positive skewness has greater deviation on the high side of the mean than on the low side; negative skewness is just the opposite.

Although skewness is related to tail imbalance, kurtosis is related to the end of the tail, regardless of whether they are above or below the mean. The fine kurtosis distribution has positive hyperkurtosis and has more extreme (at either tail) data values (for example, five or more standard deviations from the mean) than predicted by the normal distribution. Negative hyperkurtosis, called platykurtosis, is characterized by a distribution with extreme value characteristics that is less extreme than a normal distribution.

For example, as an application of skewness and kurtosis, the analysis of fixed income securities requires careful statistical analysis to determine the volatility of the portfolio when interest rates change. The model that predicts the direction of movement must consider skewness and kurtosis to predict the performance of the bond portfolio. These statistical concepts can be further applied to determine the price changes of many other financial instruments (such as stocks, options, and currency pairs).

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