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RSI or relative strength index is an economic indicator that trader use, with a number of other financial indicators, to determine whether a financial instrument is pushing closer to an upswing in a stock price or is more likely to begin, or continue, dropping in value.  
The RSI is calculated by using the following formula:
RSI = 100 – 100/(1+RS)
RS = Average of days that the stock has closed up / average of days the stock closes down
In the previous formula the higher the RSI the more likely the financial instrument will experience a downturn.  In contrast, the lower the RSI the more likely that the financial instrument is ready to start, or continue an upswing.  Usually an RSI value of 70 or more is an indicator that the financial instrument is more likely to take a downturn.  In the opposite, an RSI  of around 20 is an indicator that a stock is going to begin an upswing in value.
An example of how RSI is calculated is the following:  
Stock A has been closed up 80 out of the last 100 days.  In the RS calculation the formula will be 80/20, which will be 4.  Now that that has been calculated the RSI can be determined by the following:
100 – 100/(1+4)=RSI
100 – 100/5=RSI
High RSI numbers are an indicator of possible downward movement due to the idea that if a stock tends to increase exponentially it will eventually hit a resistance level and the stock will beg in to fall.  In contrast, an example of an RSI that will encourage an investor to buy would be one with a low RSI.  In the next example the number of days where the stock closed up will be 20 and the amount of days that the stock closed down is 80.  First we will calculate the Relative Strength:
RS = 20/80
RS = .25
Now that the RS is calculated we can calculate the RSI:
RSI = 100 – 100/(1+RS)
RSI = 100 – 100/1.25
RSI = 100 – 80
RSI = 20
Here the low RSI is an indicator that the financial instrument will be more likely in to begin an upswing in value.