# Standard Deviation

## What is 'Standard Deviation'

Standard deviation is a measure of the dispersion of a set of data from its mean; more spread-apart data has a higher deviation. Standard deviation is calculated as the square root of variance. In finance, standard deviation is applied to the annual rate of return of an investment to measure the investment's volatility.

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## BREAKING DOWN 'Standard Deviation'

Standard deviation is a statistical measurement that sheds light on historical volatility. For example, a volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is lower. A large dispersion indicates how much the return on the fund is deviating from the expected normal returns.

Investment firms report the standard deviation of their mutual funds and other products. In the finance industry, standard deviation is one of the key fundamental risk measures that analysts, portfolio managers, wealth advisors and financial planners use. Also, because it is easy to understand, this statistic is often reported to the end clients and investors on a regular basis.

## Standard Deviation Calculation

The formula for standard deviation uses three variables:

x = each individual data point

M = the mean, or average, of the data points

n = the number of data points

Using these variables, the formula for standard deviation is the square root of the sum of (x - M) ^2 / n.

Consider the following eight data points: 3, 5, 6, 4, 4, 1, 2, 7. The average of these data points is (3 + 5 + 6 + 4 + 4 + 1 + 2 + 7) / 8 = 4.

Next, take each individual data point, subtract the mean and square the results:

Data Point 1: (3 - 4) ^ 2 = 1

Data Point 2: (5 - 4) ^ 2 = 1

Data Point 3: (6 - 4) ^ 2 = 4

Data Point 4: (4 - 4) ^ 2 = 0

Data Point 5: (4 - 4) ^ 2 = 0

Data Point 6: (1 - 4) ^ 2 = 9

Data Point 7: (2 - 4) ^ 2 = 4

Data Point 8: (7 - 4) ^ 2 = 9

Add up these results and divide the sum by the number or data points â€“ in this case, 8:

Sum of squares = (1 + 1 + 4 + 0 + 0 + 9 + 4 + 9) / 8 = 3.5

Finally, to find the standard deviation, take the square root of this number:

Square Root(3.5) = 1.87

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