Overview
Sto.chas.tic (sto kas'tik) adj. 2. Math. designating a process having an infinite progression of jointly distributed random variables.
 Webster's New World Dictionary
The Stochastic Oscillator compares where a security's price closed relative to its price range over a given time period.
Interpretation
The Stochastic Oscillator is displayed as two lines. The main line is called "%K." The second line, called "%D," is a moving average of %K. The %K line is usually displayed as a solid line and the %D line is usually displayed as a dotted line.
Buy when the Oscillator (either %K or %D) falls below a specific level (e.g., 20) and then rises above that level. Sell when the Oscillator rises above a specific level (e.g., 80) and then falls below that level.
Buy when the %K line rises above the %D line and sell when the %K line falls below the %D line.
Look for divergences. For example, where prices are making a series of new highs and the Stochastic Oscillator is failing to surpass its previous highs.
Example
The following chart shows Avon Products and its 10day Stochastic.
I drew "buy" arrows when the %K line fell below, and then rose above, the level of 20. Similarly, I drew "sell" arrows when the %K line rose above, and then fell below, the level of 80.
This next chart also shows Avon Products.
In this example I drew "buy" arrows each time the %K line rose above the %D (dotted). Similarly, "sell" arrows were drawn when the %K line fell below the %D line.
This final chart shows a divergence between the Stochastic Oscillator and prices.
This is a classic divergence where prices are headed higher, but the underlying indicator (the Stochastic Oscillator) is moving lower. When a divergence occurs between an indicator and prices, the indicator typically provides the clue as to where prices will head.
Calculation
The Stochastic Oscillator has four variables:

%K Periods. This is the number of time periods used in the stochastic calculation.

%K Slowing Periods. This value controls the internal smoothing of %K. A value of 1 is considered a fast stochastic; a value of 3 is considered a slow stochastic.

%D Periods. This is the number of time periods used when calculating a moving average of %K. The moving average is called "%D" and is usually displayed as a dotted line on top of %K.
 %D Method. The method (i.e., Exponential, Simple, Time Series, Triangular, Variable, or Weighted) that is used to calculate %D.
The formula for %K is:
For example, to calculate a 10day %K, first find the security's highesthigh and lowestlow over the last 10 days. As an example, let's assume that during the last 10 days the highesthigh was 46 and the lowestlow was 38a range of 8 points. If today's closing price was 41, %K would be calculated as:
The 37.5% in this example shows that today's close was at the level of 37.5% relative to the security's trading range over the last 10 days. If today's close was 42, the Stochastic Oscillator would be 50%. This would mean that that the security closed today at 50%, or the midpoint, of its 10day trading range.
The above example used a %K Slowing Period of 1day (no slowing). If you use a value greater than one, you average the highesthigh and the lowestlow over the number of %K Slowing Periods before performing the division.
A moving average of %K is then calculated using the number of time periods specified in the %D Periods. This moving average is called %D.
The Stochastic Oscillator always ranges between 0% and 100%. A reading of 0% shows that the security's close was the lowest price that the security has traded during the preceding xtime periods. A reading of 100% shows that the security's close was the highest price that the security has traded during the preceding xtime periods.
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Information for this article was derived from the book, Technical Analysis from A to Z by Steven B. Achelis