What is the Normal Distribution
The 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.
BREAKING DOWN 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 that the variables are sampled from (provided it has finite variance).
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.
While 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.