# Standard Deviation

## Definition and Usage in Stocks and Shares

#### Standard Deviation Definition

Standard deviation is signified in mathematics by the sign σ which is lower-case Sigma, the 18th letter in the Greek alphabet. Its definition is that it is a measure of statistical dispersion, indicating how far from the mean (ie average) value typical individual members of a sample generally are. If the standard deviation is a large value, this indicates that members of the sample deviate largely from the average of tha sample, whereas a small standard deviation indicates that individual members of the sample are generally closer in value to the average.

The term Standard Deviation was first used by Karl Pearson Pearson in a lecture on 31 January 1893, although the idea was already almost a century old by then. Pearson went on to write about Standard Deviation in his book, "Contributions to the Mathematical Theory of Evolution" in 1894. He was a significant contributor in the first serious development of statistics as a scientific subject, and founded the Department of Applied Statistics in 1911 at University College in London, England. This was the world's first statistics department in a university. Much of Pearson's thinking still underpins many classical statistical methods which remain in common use today.

The concept of standard deviation can probably be best illustrated by an example - you can see a step-by-step calculation at Pandacash's Calculating Standard Deviation page.

#### Usage of Standard Deviation in Stock and Share Technical Analysis

When used in stock and share graphs, standard deviation bands are plotted above and below the simple moving average of the same period as the standard deviation period. For example, when plotting 100 day standard deviation bands, firstly the 100 day simple moving average is calculated and plotted. Then, having calculated the 100 day standard deviation, the value of the standard deviation is added to the value of the moving average, and plotted as today's point for the top Standard Deviation band. Similarly, for the bottom band the standard deviation value is deducted from the moving average and plotted as the point for the bottom band.

In many cases, instead of the usage of just a single standard deviation, multiples of the standard deviation are used as the top and bottom band lines. There is good mathematical logic behind this. Empirical mathematical rules state that for a Normal Distribution, 67% of the data in a normally distributed dataset is contained within one standard deviation of the average. 95.4 percent of the data is contained within 2 multiples of the standard deviation away from the average, and 99.7 percent of the data falls within 3 standard deviations. Whilst it is arguable whether a set of stock prices forms a true Normal Distribution, the basic idea of using standard deviations to indicate extremes of high and low points is considered among analysts to be a useful practice.

Standard Deviation bands may be based on a moving average of any period - one the most common periods used is a 20 day moving average, with two standard deviation bands as a signal line. However, for short term trading, a 10 day moving average with a 1.5 multiple of the standard deviation can be used, or for longer term strategies, a 50-day moving average with a signal line of 2.5 standard deviations may be plotted.

Due to the fact that the moving average, and the Standard Deviation is calculated each day (or hour if hourly data is used), different values for the standard deviation exist each day (or hour). This results in the width between the upper and lower Standard deviation bands varying as time goes by and new calculations are done. Thus, the bands are "self-adjusting", expanding when the volatility over the period under study is high, and contracting when the volatility decreases.

Volatile stocks, shares and other securities are considered to be high-risk and high-reward securities. If your investment strategy leans specifically toward an interest in investing in low risk stocks, you may consider investing in a stock which, over a long term, has a consistently small bandwidth (difference between the top and bottom Standard Deviation bands). Conversely, if you have a higher risk/reward strategy, then stocks which have a consistently wide bandwidth would be the type of investment you may be looking for.

Volatility is also used in option prices, (including "Traded Options" in the UK). A security with a higher volatily will have a higher time value part of the option premium, whereas is volatility is low, the time value of the option premium will be lower. This is because a stock with higher volatility would be expected to exhibit larger price fluctuations. Therefore knowing the security's current and historical bandwidth may help in your decision regarding whether you wish to buy an option at the current option price.

Standard deviations can also be used to identify extreme stock and share values, thus acting as an overbought or oversold indicator. For example, technical analysis practitioners may consider that if a stock price is beyond the top 2 standard deviations band, that stock could well be in overbought territory, as assuming a "Normal Distribution" of stock prices, the stock would be expected to only reach such a level only only 4.6% of the time (because, as mentioned above, 95.4% of the price data should be contained within 2 multiples of the standard deviation assuming a Normal Distribution). An even stronger overbought signal would be if the stock reached beyond 3 multiples of the standard deviation, as this would only be expected to happen only 0.27 per cent of the time.

Similar to most overbought or oversold indicators, Standard Deviations should be considered secondary signals rather than primary identifiers of whether to make a purchase or sale. They shouldn't be used on their own as the basis of a trading decision. This is mainly because sometimes, once a stock has reached the two or three standard deviation band, it may hug the band for some time, rather than bounce back off it. This can happen, for example, in massive buying rallies or selling periods where the actions of investors continue to push the price in one direction beyond expectations. Standard deviations should be thought of as a measure of how extreme and abnormal the current price of a stock or share is in comparison to its recent history. The more extreme a price is away from its average the more likely it is that the price will head toward the average soon. Of course, bear in mind that moving averages, by their name and nature, move themselves - so it can somtimes be that the stock price may remain around the same level, but the moving average could come to meet it rather than the stock price moving to meet the moving average.

Whilst maintaining attention to these cautions, it is believed by market analysts that prices abhor extremes and, over time, return to average levels. If a the price of a security is at or very closely approching the two or three standard deviation band, it could well indicate an imminent reversal - usage of the MACD or RSI to help your buying or selling decision in conjuction with Standard Deviations is a strategy recommended by some analysts.

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