Gauss and the Bell Curve

 Karl Fredrich Gauss was a brilliant mathematician who lived in the early 1800’s and gave the world quadratic equations, methods of least squares fitting and the normal distribution. Gauss defined the normal distribution as the mean error while mathematician Karl Pearson defined it as standard deviation in the early 1900’s.
Modern day terminology defines the normal distribution as the bell curve. Ironically, Gauss intended in 1809 to answer an astronomy question not to find or understand normal distributions. Another mathematician of the era Pierre Simon LaPlace actually was the founder of the normal distribution from the paper Gauss published regarding his astronomy question in 1809.
The normal distribution was founded by sheer accident yet credited to Gauss because it appeared in print by him and has been the subject of much study by mathematicians for 200 years. The entire study of statistics originated from Gauss and thankfully so because it allowed us to understand markets, prices and probabilities among other applications. The only way to understand Gauss and the bell curve is to understand statistics. So I will build a bell curve in this article beginning with means and apply it to a trading example.
     Three methods exist to determine distributions, mean, median and mode.
Means are factored by adding all scores and dividing by the number of scores. Median is factored by adding the two middle numbers of a sample and divide by two. Mode is the most frequent of the numbers in a distribution of numbers.
The best method is to use means because it averages all numbers and is less subject to sample fluctuations. This was the Gaussian approach and his preferred method. What we are measuring here is parameters of central tendency or to answer where are our sample scores headed. To understand this, we must plot our scores beginning with 0 in the middle and plot + 1, + 2 and + 3 standard deviations on the right and -1, -2 and -3 on the left.
    So on a chart, plot the scores. What we will find here is .68 % of all scores will fall within -1 and + 1 standard deviations, 95 % fall within 2 standard deviations and 99 % fall within 3 standard deviations of the mean.  But this is not enough to tell us about the curve. We need to factor variances.
 Variance answers the question how spread out is our distribution.
It factors in possibilities why outliers may exist in our sample and helps us to understand these outliers and where they are plotted. So find the mean, subtract the mean from each score for a deviation score, square each deviation score and add all. Divide the sum by the number of scores. This is the variance that explains variability and may help to explain a hypothesis regarding the outliers.
   For standard deviation, we want to measure our spread more closely. So factor the square root of the variance.Here we will know exactly where our standard deviations will fall in relation to our total distribution.Modern day terms call this dispersion. In a Gaussian distribution, if we know the mean and the standard deviation, we can know the percentages of the scores that fall within plus or minus 1,2 or 3 standard deviations from the mean. This is called the confidence interval. This is how we know 68% of distributions fall within plus or minus 1 standard deviation, 95% within plus or minus 2 standard deviations and 99 % within plus or minus 3 standard deviations. Gauss called these probability functions.
     Notice our whole discussion so far is all about explanation of the mean and the various computations to help us explain it more closely. Once we plotted our distribution scores, we basically drew our bell curve above all the scores.Yet we can’t assume that all distributions will be perfectly normal where the mean will always equal 0 and the tails will be of equal length. So still this is not enough because we have tails on our curve that need explanation to better understand the whole curve. To do this we go to the third and fourth moments of statistics of the distribution called Skew and Kurtosis.
  Skewness of tails measures asymmetry of the distribution. A positive skew has a variance from the mean that is positive and skewed right while a negative skew has a variance from the mean skewed left. A symmetrical skew has 0 variance that forms a perfect normal distribution. Visually, when the bell curve is drawn first with a long tail, this is positive while the tail at the beginning before the bell curve is negative. If a distribution is symmetric, the sum of cubed deviations  above the mean will balance the cubed deviations below the mean.A skewed right distribution will have a skew greater than 0 while a skewed left distribution will have a skew less than 0.
  Kurtosis explains the peakedness of the distribution. High kurtosis has more peak and is less flat.
A perfectly normal distribution called mesokurtic has a kurtosis equal to 0. A positive distribution called leptokurtic with a high bell usually has a value greater than 3 while a negative platykurtic peak has kurtosis less than 3.
 Skew is more important to measure trades than kurtosis. Both are used to measure treasury auctions by the amount of bills or bonds sold to the skew to determine if the auction was successful. A successful auction would show a big bell curve with a short skew and positive kutosis.
Treasury bills and bonds is  the measure of interest rates and determines prices for many other financial instruments such as stocks, options and currency pairs. Skews are used to measure option prices by measuring implied volatilities by strike prices on an L shaped graph among other uses.
     Standard deviation measures volatility and asks the question can past returns equal future returns. Smaller standard deviations may mean less risk for a stock while higher volatility may mean a higher standard deviation.
Traders can measure closing prices from the average as it is dispersed from the mean. Dispersion would then measure the difference from actual value to average value. A larger difference between the two means a higher standard deviation and volatility. Prices that deviate far away from the mean will always revert back to the mean so traders can always take advantage of these situations. Prices that trade in a small range are always ready for a breakout.
The best technical indicator to use for standard deviation trades is Bollinger Bands because its a measure of volatility set at two standard deviations for upper and lower bands with a 20 day moving average. Double Bands is recommended with standard deviations set at 3. The Gauss Distribution was just the beginning of understanding of markets. It later led to Time Price Series and Garch Models as well as more applications of skew such as the Volatility Smile and other volatility skews.
 Brian Twomey is a currency trader and adjunct professor of Political Science at Gardner-Webb University

 

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