What is a ‘normal distribution’
Normal distribution also known as Gaussian distribution, the probability distribution that is symmetric about the mean, showing that the data is near the mean are more frequent than the data far from the average.
Breaking down the ‘normal distribution’
The normal distribution is the most common type of distribution, adopted in the technical analysis of the stock market and other types of statistical analysis. The standard normal distribution has two parameters: the mean and standard deviation. For normal distribution, 68% of observations are within +- one standard deviation, 95% within + two standard deviations and 99.7% within + three standard deviations.
While the actual data is usually not clearly normally distributed, the normal model is motivated by the Central limit Theorem, which States that the average is calculated from independent identically distributed random variables have approximately normal distribution, regardless of the type of distribution that the variables are selected from (assuming finite variance).
Skewness and Kurtosis
Real data rarely, if ever come from normal distributions. The skewness and kurtosis to determine how much the real distribution from the normal distribution. Asymmetry measures the symmetry of a distribution. The normal distribution is symmetric and has a skewness equal to zero, as in the case of all symmetric distributions. If the data distribution has a skewness less than zero, then the data distribution is skewed to the left; positive skewness means that the distribution shifted to the right. Asset prices can be modeled using a lognormal distribution, which is shifted to the right because asset prices are nonnegative, and because sometimes there are assets with extremely high prices against the majority.
The kurtosis statistic measures the tail ends of the distribution towards the tails of the normal distribution. The normal distribution has a kurtosis of three, which indicates that the distribution has neither fat nor thin tails. Therefore, if the observed data have a value greater than three, the distribution has heavy tails compared to normal distribution. If the data have a kurtosis less than three, then we say that has thin tails compared to normal distribution.
Returns of the stock market is often assumed to follow a normal distribution. However, in reality, distribution, returns, tend to have heavy tails, and hence have kurtosis more than three. Such returns usually moves more than three standard deviations beyond the mean more often than expected under the assumption of normal distribution.