Standard Deviation Definition

The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. 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 a higher deviation within the data set; thus, the more spread out the data, the higher the standard deviation.

Standard deviation is a statistical measurement in finance that, when applied to the annual rate of return of an investment, 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, which shows 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.

$\begin{aligned} &\text{Standard Deviation} = \sqrt{ \frac{\sum_{i=1}^{n}\left(x_i - \overline{x}\right)^2} {n-1} }\\ &\textbf{where:}\\ &x_i = \text{Value of the } i^{th} \text{ point in the data set}\\ &\overline{x}= \text{The mean value of the data set}\\ &n = \text{The number of data points in the data set} \end{aligned}$​Standard Deviation=n−1∑i=1n​(xi​−x)2​​where:xi​=Value of the ith point in the data setx=The mean value of the data set​

Standard deviation is calculated as:

1. The mean value is calculated by adding all the data points and dividing by the number of data points.
2. The variance for each data point is calculated, first by subtracting the value of the data point from the mean. Each of those resulting values is then squared and the results summed. The result is then divided by the number of data points less one.
3. The square root of the variance—result from no. 2—is then taken to find the standard deviation.

For an in-depth look, read more about calculating standard deviation and other volatility measures in Excel.

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, one can expect an index fund to have a low standard deviation versus its benchmark index, as the fund's goal is to replicate the index.

On the other hand, one can expect aggressive growth funds to have a high standard deviation from relative stock indices, as their portfolio managers make aggressive bets to generate higher-than-average returns.

A lower standard deviation isn't necessarily preferable. It all depends on the investments one is making, and one's willingness to assume the risk. When dealing with the amount of deviation in their portfolios, investors should consider 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.

Standard deviation is one of the key fundamental risk measures that analysts, portfolio managers, advisors use. Investment firms report the standard deviation of their mutual funds and other products. A large dispersion shows how much the return on the fund is deviating from the expected normal returns. Because it is easy to understand, this statistic is regularly reported to the end clients and investors.

Say we have the data points 5, 7, 3, and 7, which total 22. You would then divide 22 by the number of data points, in this case, four—resulting in a mean of 5.5. This leads to the following determinations: x̄ = 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 is 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 minus 1, which is 3, resulting in a variance approximately of 3.67.

The square root of the variance is then calculated, which results in a standard deviation measure of approximately 1.915.

Or consider shares of Apple (NASDAQ: AAPL) for the last five years. Returns for Apple’s stock were 37.7% for 2014, -4.6% for 2015, 10% for 2016, 46.1% for 2017 and -6.8% for 2018. The average return over the five years is 16.5%.

The value of each year's return less the mean is 21.2%, -21.2%, -6.5%, 29.6%, and -23.3%. All those values are then squared to yield 449.4, 449.4, 42.3, 876.2, and 542.9, respectively. The variance is 590.1, where the squared values are added together and divided by 4 (N minus 1). The square root of the variance is taken to get the standard deviation of 24.3%.

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 the square root of the 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 dataset.

Standard deviations are usually easier to picture and apply. The standard deviation is expressed in the same unit of measurement as the data, which isn't necessarily 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.

The biggest drawback of using standard deviation is that it can be impacted by outliers and extreme values. Standard deviation assumes a normal distribution and calculates all uncertainty as risk, even when it’s in the investor's favor—such as above average returns.