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What is the 'Normal Distribution'

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is a symmetric about the mean, showing that data near the mean are more frequent 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 deviations of the mean, 95% are within +- two standard deviations, and 99.7% are within +- three standard deviations.

While real data are usually not precisely normally distributed, the normal model is motivated by the Central Limit Theorem, which 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 data rarely if ever come from normal distributions. The skewness and kurtosis coefficients measure how different the real 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, as is the case with all symmetric distributions. If the distribution of a data set has a skewness less than zero, the distribution of the data is skewed to the left; positive skewness implies that the distribution is skewed to the right. Asset prices can be modelled using a lognormal distribution, which is skewed to the right because asset prices are non-negative, and because there are occasional assets with extremely high prices relative to the majority.

The kurtosis statistic measures the tail ends of a distribution in relation to 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 observed data have a kurtosis greater than three, the distribution is said to have heavy tails when compared to the normal distribution. If the data have a kurtosis less than three, it is said to have thin tails when compared to the normal distribution.

Stock market returns are often assumed to follow a normal distribution. However, in reality, return distributions tend to have fat tails, and therefore have kurtosis greater than three. Such returns have typically had moves greater than three standard deviations beyond the mean more often than expected under the assumption of a normal distribution.

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