Normal Distribution

What is Normal Distribution?

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

Understanding Normal Distribution

The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses. The standard normal distribution has two parameters: the mean and the standard deviation. For a normal distribution, 68% of the observations are within +/- one standard deviation of the mean, 95% are within +/- two standard deviations, and 99.7% are within +- three standard deviations.

The normal distribution model is motivated by the Central Limit Theorem. This theory states that averages calculated from independent, identically distributed random variables have approximately normal distributions, regardless of the type of distribution from which the variables are sampled (provided it has finite variance). Normal distribution is sometimes confused with symmetrical distribution. Symmetrical distribution is one where a dividing line produces two mirror images, but the actual data could be two humps or a series of hills in addition to the bell curve that indicates a normal distribution.

Skewness and Kurtosis

Real life data rarely, if ever, follow a perfect normal distribution. The skewness and kurtosis coefficients measure how different a given distribution is from a normal distribution. The skewness measures the symmetry of a distribution. The normal distribution is symmetric and has a skewness of zero. If the distribution of a data set has a skewness less than zero, or negative skewness, then the left tail of the distribution is longer than the right tail; positive skewness implies that the right tail of the distribution is longer than the left.

The kurtosis statistic measures the thickness of the tail ends of a distribution in relation to the tails of the normal distribution. Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean). Distributions with low kurtosis exhibit tail data that is generally less extreme than the tails of the normal distribution. The normal distribution has a kurtosis of three, which indicates the distribution has neither fat nor thin tails. Therefore, if an observed distribution has a kurtosis greater than three, the distribution is said to have heavy tails when compared to the normal distribution. If the distribution has a kurtosis of less than three, it is said to have thin tails when compared to the normal distribution.

How Normal Distribution is Used in Finance

The assumption of a normal distribution is applied to asset prices as well as price action. Traders may plot price points over time to fit recent price action into a normal distribution. The further price action moves from the mean, in this case, the more likelihood that an asset is being over or undervalued. Traders can use the standard deviations to suggest potential trades. This type of trading is generally done on very short time frames as larger timescales make it much harder to pick entry and exit points.

Similarly, many statistical theories attempt to model asset prices under the assumption that they follow a normal distribution. In reality, price distributions tend to have fat tails and, therefore, have kurtosis greater than three. Such assets have had price movements greater than three standard deviations beyond the mean more often than would be expected under the assumption of a normal distribution. Even if an asset has went through a long period where it fits a normal distribution, there is no guarantee that the past performance truly informs the future prospects.