Normal Distribution

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What is the 'Normal Distribution'

The normal distribution, also known as the Gaussian or standard normal distribution, is the probability distribution that plots all of its values in a symmetrical fashion, and most of the results are situated around the probability's mean. Values are equally likely to plot either above or below the mean. Grouping takes place at values close to the mean and then tails off symmetrically away from the mean.

BREAKING DOWN 'Normal Distribution'

The normal distribution is the most common type of distribution and is often found in stock market analysis. Given enough observations within a sample size, it is reasonable to make the assumption that returns follow a normally distributed pattern, but this assumption can be disproved.

As with any distribution, the distribution's mean, skewness and kurtosis coefficients should be calculated to determine the type of distribution. The standard normal distribution has two parameters: the mean and the standard deviation. In the Gaussian distribution, the mean, or mu, is equal to zero, while the standard deviation, or sigma, is equal to one. Under normal circumstances, independent identically distributed random variables, or outcomes, are said to converge to a standard normal distribution under the central limit theorem.

Skewness and Kurtosis

The skewness measures the symmetry of a distribution. The standard normal distribution has a skewness of zero, and therefore, it is said to be symmetric. If the distribution of a data set has a skewness less than zero, the data is skewed to the left. Conversely, data that has a positive skewness is said to be skewed to the right. For example, asset prices follow a lognormal distribution, which is skewed to the right because asset prices are non-negative.

The kurtosis measures the tail ends of a distribution and whether the distribution of a data set has skinny tails or fat tails in relation to the normal distribution. The standard normal distribution has a kurtosis of three, which indicates data that follow a Gaussian distribution have neither fat or thin tails. Therefore, if observed data have a kurtosis greater than three, it 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.

For example, stock market returns are said to follow a normal distribution in theory. However, in reality, asset returns tend to have fat tails. Observed asset returns have typically had moves greater than three standard deviations beyond the mean more than expected under the assumptions of the normal distribution.