The main difference between fast and slow stochastics is summed up in one word: sensitivity. The fast stochastic is more sensitive than the slow stochastic to changes in the price of the underlying security and will likely result in many transaction signals. However, to really understand this difference, you should first understand what the stochastic momentum indicator is all about.
The stochastic momentum oscillator is used to compare where a security's price closed relative to its price range over a given period of time. It is calculated using the following formula:
%K = 100[(C â€“ L14)/(H14 â€“ L14)] 
C = the most recent closing price
L14 = the low of the 14 previous trading sessions
H14 = the highest price traded during the same 14day period.
A %K result of 80 is interpreted to mean that the price of the security closed above 80% of all prior closing prices that have occurred over the past 14 days. The main assumption is that a security's price will trade at the top of the range in a major uptrend. A threeperiod moving average of the %K called %D is usually included to act as a signal line. Transaction signals are usually made when the %K crosses through the %D. Generally, a period of 14 days is used in the above calculation, but this period is often modified by traders to make this indicator more or less sensitive to movements in the price of the underlying asset. The result obtained from applying the formula above is known as the fast stochastic. Some traders find that this indicator is too responsive to price changes, which ultimately leads to being taken out of positions prematurely. To solve this problem, the slow stochastic was invented by applying a threeperiod moving average to the %K of the fast calculation. Taking a threeperiod moving average of the fast stochastic's %K has proved to be an effective way to increase the quality of transaction signals; it also reduces the number of false crossovers. After the first moving average is applied to the fast stochastic's %K, an additional threeperiod moving average is then applied  making what is known as the slow stochastic's %D. Close inspection will reveal that the %K of the slow stochastic is the same as the %D (signal line) on the fast stochastic.
An easy way to remember the difference between the two is to think of the fast stochastic as a sports car and the slow stochastic as a limousine. Like a sports car, the fast stochastic is agile and changes direction very quickly in response to sudden changes. The slow stochastic takes a little more time to change direction. Mathematically, the two oscillators are nearly the same except that the slow stochastic's %K is created by taking a threeperiod average of the fast stochastic's %K. Taking a threeperiod moving average of each %K will result in the line that is used for a signal.
For more insight, read Getting To Know Oscillators  Part 3: Stochastics or Support, Resistance, Stochastics, and EMA

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