Investopedia: What is 'Standard Deviation'?

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Standard Deviation
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What is 'Standard Deviation'

Standard deviation is a measure of the dispersion of a set of data from its mean. It is calculated as the square root of variance by determining the variation between each data point relative to the mean. If the data points are further from the mean, there is higher deviation within the data set.

In finance, standard deviation is a statistical measurement; when applied to the annual rate of return of an investment, it sheds light on the historical volatility of that investment. The greater the standard deviation of a security, the greater the variance between each price and the mean, indicating a larger price range. For example, a volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is usually rather low.
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BREAKING DOWN 'Standard Deviation'

In the financial services industry, standard deviation is one of the key fundamental risk measures that analysts, portfolio managers, wealth management advisors and financial planners use. Investment firms report the standard deviation of their mutual funds and other products. A large dispersion indicates how much the return on the fund is deviating from the expected normal returns. Because it is easy to understand, this statistic is often reported to the end clients and investors on a regular basis.
What's the Difference Between Standard Deviation and Mean?

In its simplest form, the mean is simply the average of all the data points in a given set. In investing, for example, you might want to know the mean closing price for the last 20 days. This can be obtained by adding the closing prices for each session and dividing by 20. Because markets are fickle at best, traders and analysts use moving averages that adjust daily to incorporate the most updated data. This means the calculation is always taking into account the most recent sessions' movements, and older sessions drop off as they become less relevant. An exponential moving average (EMA) is calculated by weighting each data point, giving greater significance to more recent data.

Standard deviation is calculated based on the mean. The distance of each data point from the mean is squared, summed and averaged to find the variance. Or to put it another way: Variance is derived by taking the mean of the data points, subtracting the mean from each data point individually, squaring each of these results and then taking another mean of these squares. Standard deviation is simply the square root of the variance.
Calculating a Standard Deviation

The formula for standard deviation uses three variables. The first variable is to be the value of each point within the data set, traditionally listed as x, with a sub-number denoting each additional variable (x, x1, x2, x3, etc.). The mean, or average, of the data points is applied to the value of the variable M, and the number of data points involved is assigned to the variable n.

To determine the mean value, the values of the data points must be added together, and that total is then divided by the number of data points that were included. For example, if the data points were 5, 7, 3 and 7, the total would be 22. That total of 22 would then be divided by the number of data points, in this case four, resulting in a mean of 5.5. This leads to the following determinations: M = 5.5 and n = 4.

The variance is determined by subtracting the value of the mean from each data point, resulting in -0.5, 1.5, -2.5 and 1.5. Each of those values are then squared, resulting in 0.25, 2.25, 6.25 and 2.25. The square values are then added together, resulting in a total of 11, which is then divided by the value of n-1, which is 3 in this instance, resulting in a variance approximately of 3.67.

The square root of the variance is then calculated, resulting in the standard deviation of approximately 1.915.
Standard Deviation vs. Variance

The variance helps determine the data's spread size when compared to the mean value. As the variance gets bigger, more variation in data values occurs, and there may be a larger gap between one data value and another. If the data values are all close together, the variance will be smaller. This is more difficult to grasp than are standard deviations, however, because variances represent a squared result that may not be meaningfully expressed on the same graph as the original data set.

Standard deviations are usually easier to picture and apply. The standard deviation is expressed the same unit of measurement as the data, which isn't necessarily be the case with the variance. Using the standard deviation, statisticians may determine if the data has a normal curve or other mathematical relationship. If the data behaves in a normal curve, then 68% of the data points will fall within one standard deviation of the average, or mean data point. Bigger variances cause more data points to fall outside the standard deviation. Smaller variances result in more data that is close to average.
What's Standard Deviance Used For?

Standard deviation is an especially useful tool in investing and trading strategies, as it helps measure market and security volatility – and predict performance trends.

As it relates to investing, for example, an index fund can be expected to have a low standard deviation versus its benchmark index, as the fund's goal is to replicate the index. Aggressive growth funds, on the other hand, can be expected to have a high standard deviation from relative stock indices, as their portfolio managers make aggressive bets in an effort to generate higher-than-average returns.

A lower standard deviation isn't necessarily preferable. It all depends on the sort of investments one is making, and the one's willingness to assume risk. When dealing with the amount of deviation in their portfolios, investors should consider both their personal tolerance for volatility and their overall investment objectives. More aggressive investors may be comfortable with an investment strategy that opts for vehicles with higher-than-average volatility, while more conservative investors may not.


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