Mathematical Formula for Standard Deviation

Standard Deviations

Standard deviation is simply a measure of distribution. It shows how data is clustered around the average value for a given set of data. For a given set of data, a one standard deviation move above and below the average encompasses 2/3rds of the data under consideration. A two standard deviation move above and below the average includes 96% of the data and a three standard deviation move above and below the average includes 99% of the data.

In Using too much Leverage discussed measuring standard deviations of monthly returns in order to arrive at a conservative account size for a given portfolio. The following formula can be used to calculate the standard deviation of any group of data. The first step is to calculate the "average" value. The second step measures the deviations from the average to arrive at the standard deviation.

• Step 1 — Calculate the Average Value over the Number of Periods under Consideration
  This is accomplished by adding up the sum of all of the values and dividing by the number of periods:
  sum1 = 0
  For x = 1 to Number of Periods
  sum1 = sum1 + value(x)
  Next x
  average = (sum1 / Number of Periods)
• Step 2 — Calculate the Deviation from the Average for Each Value
• Step 3 — Calculate the Standard Deviation Steps 2 and 3 are done by:
  a) Subtracting each value from the average.
  b) Squaring "a" (simply multiply "a" times "a").
  c) Sum the value for "b" for each value.
  d) Take the square root of ("c" divided by Number of Periods).
  sum2 = 0
  For x = 1 To Number of Periods
  sum2 = sum2 + ((average, value(x) x (average, value(x))
  Next x
  Standard Deviation = Square Root(sum2 / Number of Periods)

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