A trader does not have to be a statistical genius, but he or she should have a basic understanding of statistics that are descriptive of various properties of the data being analyzed. This article gives an overview of some of the most important statistical concepts that traders should understand. These concepts are as follows:
Mean, median, and mode.
Standard deviation.
Types of distributions and their properties.
How mean and standard deviation interact.
Hypothesis testing.
Mean or variance with two or more distributions.
Linear correlation.
A trader who has a general idea of these concepts can use statistics to develop trading systems as well as to test patterns and relationships. Let's now discuss each of these concepts in more detail.
Mean, Median, and Mode
The mean is another term for the average. The median of a sample is the middle value, based on order of magnitude. For example, in the number sequence 1,2,3,4,5,6,7,8,9, the median is 5 because it is surrounded by four higher values and four lower values. The mode is the most frequently occurring element.
To clarify the definitions of mean, median, and mode, let's look at two different cases:
1,2,3,4,5,6,7,8,9,10.
1,2,3,4,5,100,150,200,300.
In the first case, the mean is 5.5 and the median is either 5 or 6. Hence, in this case, the mean and the median are similar. In the second case, the median would still be 5, but the mean would be 85. In what is called a normal distribution, the mean and median are similar. When the distribution is not normal, the mean and the median can be different.
Types of Distributions and Their Properties
Most standard statistical methods are based on what is called a normal or Gaussian distribution. This is the standard bell curve, which is represented in the symmetrical chart shown below.
A distribution can have several different properties. The first is skewness. The skewness of a distribution is the degree of its asymmetry around the mean. Positive skewness indicates a distribution with an asymmetric tail extending more toward positive values. Negative skewness indicates a distribution with an asymmetric tail extending more towards negative values. Next, we observe the kurtosis of the distribution. Kurtosis characterizes the relative peakedness or flatness of a distribution, compared to the normal distribution. A negative kurtosis indicates a relatively flat distribution. When we look at the distribution of financial data, we see some interesting things. First, financial data have a distribution that is leptokurtotic: Large moves occur more than they should for a normal distribution. This property of being leptokurtotic is very important. The fact that large moves occur more than they should is why trend-following systems work.
The chart below shows the distribution of 5-day returns for the D-Mark from 2/13/75 to 7/1/96. Dr. Benoit Mandelbrot, one of the patriarchs of chaos theory, suggested in 1964 that the capital markets follow a family of distributions he called "stable Paretian." Stable Paretian distributions have higher peaks at the mean and fat tails, and they are characterized by a tendency to trend as well as to be cyclical.
They also have discontinuous changes, and they can be adjusted for skewness. They are different from simple leptokurtotic gaussian distributions in that they have an infinite or undefined variance. These types of distributions are now called "fractal distributions." Because financial markets do not follow a gaussian distribution, using standard statistics can give us only an imperfect estimate, but that estimate is good enough to give us an edge.
The Concept of Variance and Standard Deviation
The variance and standard deviation of a data set are very important. The variance is the average deviation of the data set from its mean. The variance is calculated as follows:
Where N is the number of elements, M is the same mean, and D is the current value. The standard deviation is simply the square root of the variance. The standard deviation has some very interesting trading applications and is used in many different indicators. A Bollinger band, for example, is simply a price band drawn two standard deviations away from the mean.
How Gaussian Distribution, Mean, and Standard Deviation Interrelate
The interaction between the mean and the standard deviation has many trading applications. First, there is a basic relationship between the mean of a data set and the standard deviation. This relationship states that, for a normal or standard distribution, 68 percent of the data is contained within one standard deviation and 95 percent of the data is contained within two standard deviations. Almost all of the data is contained within three standard deviations. For a normal distribution, this number is 95.5 percent.
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