## Technical indicators: A statistical approach

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The practitioner of technical analysis views markets based solely on price and volume data. Consistent with this approach, the technical analyst seeks tools that are independent of any particular market or time frame. The goal is to trade any market and time frame relying on the best set of technical indicators available. For example, a reliable trending indicator plus volume study should be applicable whether the asset traded is S&P 500 futures or INTC (Intel Corp.) stock. Technical indicators that are market-independent are said to be robust.

Some of the best known technical indicators are asset-specific and may not be robust. For example, the well-known and widely used moving average convergence-divergence (MACD) indicator outputs the difference between two exponential moving averages, and the moving averages are based on the price of the underlying asset. In this case, using similar five-minute bar charts, an overbought/oversold (OB/OS) MACD level might be eight ticks (two points) for the S&P 500 E-mini (ES) contract and five ticks ($0.05) for INTC shares. Using the MACD across a variety of assets and timeframes requires that the trader figure OB/OS levels for each application, which can be time-consuming and error-prone.

Here, we’ll describe a simple method of making these technical indicators robust. It converts indicator outputs from asset-specific values to statistical measures of price extension and compression. This results in robust indicators that can be used across markets without modification.

**Statistical indicators**

The most common method of estimating the spread, or dispersion, of a data set is standard deviation. The data set’s mean, or average, is first calculated. The Greek letter μ (“mu”) stands for the data set’s mean value. The data set’s standard deviation is then calculated as the average distance of the data points from their mean. The standard deviation, referred to with the Greek letter σ (“sigma”), is easy to work with because it takes values that are the same units as the original underlying data.

The science of statistics has determined that for many naturally occurring populations (population height, weight, test scores, etc.), data are “normally” distributed about their mean. This is the well-known bell curve of population distribution. Interestingly, bell curves are completely defined by their mean and standard deviation. This allows us to say that a normal distribution has approximately 70% of its data contained with one standard deviation of its mean and 95% within two standard deviations, regardless of the values computed for the data set’s mean and standard deviation. For example, if test scores range from 0 to 100, with a mean of 75 and a standard deviation of 10, then we can predict that 70% of the scores range from 65 to 85 (75 - 10 to 75 + 10).

It is questionable whether financial markets can be accurately modeled using a normal distribution. Markets often exhibit “fat tails,” meaning there may be an unexpected amount of data far from the mean, say outside of the two, or even three, standard deviation levels. Fat tails model market panics and over-exuberance, and there is a large body of financial analysis that uses different assumptions of dispersion than the normal distribution. Nonetheless, normal distribution works as a good first approximation that can be used by traders. For example, the Market Profile Value Area is defined as encompassing 70% of price movement beginning from a mode (most common) price. The 70% value represents one standard deviation variation in price.

Making use of standard deviation with market price and, in particular, bar charts, is straightforward. The standard deviation of recent price history is first calculated using some number of previous bar prices. Picking a history length is similar to selecting the length for a moving average indicator. Then a number of standard deviations can be assigned to recent price action. Specifically, the current price, **x**, is said to be at “**(x Ð μ)/σ** standard deviations;” that is, the current price, **x**, is some number of standard deviations displaced from the mean.

The value **(x Ð μ)/σ** may at first seem obscure. But consider the meaning of any fraction or ratio; for example, the fraction one-third. A useful interpretation of the fraction one-third asks, how many threes are there in one? In the same way, the ratio **(x Ð μ)/σ** asks, how many standard deviations are there in **x Ð μ,** the distance of the current price from market mean price? The technical term for the ratio **(x Ð μ)/σ** is “z-score.” By converting indicator output values to z-scores, we are able to move technical indicators to a statistical basis.