Normal distribution

Normal distribution is a bell shaped curve, symmetrical around the mean of the sample analyzed. The normal Distribution is the most commonly probability distribution in the investment industry.

Normal Distribution

Normal distribution is characterized by the follow key statistics:
Mean: This is the average daily return from the sample analyzed. It is also known as the expected return which is the amount we expect to make if we buy on the open and sell on the close on any given day.
Median: This is the middle number of a sample. When median is greater than mean, it implies that negative returns must be higher than positive ones.
Standard deviation: A sample described from a normal distribution means that 68.2% of data lies within 1 standard deviation either side of the mean, 95.4% lies within 2 and 99.8% within 3 standard deviation.
Kurtosis: Kurtosis is used to describe the distribution of observed data around the mean. It is the volatility of volatility. A high kurtosis is a distribution with fat tails and a low kurtosis characterizes a distribution with skinny tails and concentrated towards the mean. A high Kurtosis tells us that there is greater chance of extreme outcomes than a bell curve of normal distribution would suggest.
Skewness: Skewness descsribes the asymmetry from the normal distribution in a set of data. Skewness is important to investing. Most sets of asset returns, have either positive or negative skew rather than following a balance normal distribution of zero skew. A high Skewness means that if the market goes down, the value of an asset also goes down to an even greater degree.
Financial returns exhibit fat tails. As a result the tail risk for a specific investment is much higher than predicted by the simple model of normal distribution.
Human Height Example:
A sample of 500 people was taken and their height was measured.
Gaussian distribution

The mean of the data set was measured as 180cm, with a standard deviation of 10cm. A standardized normal distribution can be plotted over top of this histogram to access its “goodness of fit”.
The normal distribution (in blue) describes fairly well the actual sample data (in red). This chart has been standardized and displayed in terms of standard deviations. 0 standard deviations from mean represent the height mean value itself in this case 180cm whereas plus 1 and -1 standar deviation represent plus (or minus) 10cm which is equal to 1 standard deviation.
In a perfect normal distribution, standard deviations from the mean account for the following number of occurrences:

Standard Deviations from the mean Percentage of occurrences in data set
1 68.2%
2 95.4%
3 99.8%

In this case 95.4% of values of a sample lie between +2 and -2 standard deviations from the mean. In the specific example 95.4% of the people will lie between 160cm-200cm.