Loading the player...

What is 'Standard Deviation'

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 higher deviation within the data set; thus, the more spread out the data, the higher the standard deviation.

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, 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.

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 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. 

What's the Difference Between Standard Deviation and Mean?

In its simplest form, the mean is the average of all the data points in a set. In investing, for example, you might want to know the mean closing price for the last 20 days. You can find this by adding the closing prices for each session and dividing by 20. Because markets are fickle, traders and analysts use moving averages that adjust daily to incorporate the most recent data. This means the calculation is always considering the most recent sessions' movements, and older sessions drop off as they become less relevant. One can calculate an exponential moving average 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 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, you must add the values of the data points together, and then divide that total by the number of data points included. For example, if the data points were 5, 7, 3 and 7, the total would be 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: 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, which results in a standard deviation measure 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 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 percent 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, 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 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.

RELATED TERMS
  1. Variability

    Variability is the extent to which data points in a statistical ...
  2. Empirical Rule

    The empirical rule is a statistical rule stating that for a normal ...
  3. Yield Variance

    Yield variance describes the difference between actual output ...
  4. Variance Swap

    A variance swap allows counterparties to hedge or speculate directly ...
  5. Budget Variance

    A budget variance is a measure used to quantify the difference ...
  6. Sales Price Variance

    Sales price variance is the difference between the money a business ...
Related Articles
  1. Trading

    Improve your investing with Excel

    Find out how to use Excel, a useful tool for assisting with investment organizations and evaluations.
  2. Trading

    Exploring the Exponentially Weighted Moving Average

    Learn how to calculate a metric that improves on simple variance.
  3. Trading

    The Linear Regression of Time and Price

    This investment strategy can help investors be successful by identifying price trends while eliminating human bias.
  4. Investing

    Stock Market Risk: Wagging The Tails

    The bell curve is an excellent way to evaluate stock market risk over the long term.
  5. Investing

    Understanding Volatility Measurements

    Learn how to choose a fund with an optimal risk-reward combination. Find more information about standard deviation, beta, and more.
  6. Investing

    Two Approaches to Building a Low-Risk Portfolio

    Building a portfolio consisting of low-risk assets is achieved primarily by using one of two principal low-volatility strategies.
  7. Trading

    The Normal Distribution Table, Explained

    The normal distribution formula is based on two simple parameters - mean and standard deviation
  8. Investing

    How Investment Risk Is Quantified

    FInancial advisors and wealth management firms use a variety of tools based in modern portfolio theory to quantify investment risk.
  9. Investing

    XLI Vs. VIS: Comparing Industrials ETFs

    Learn about the industrials sector, how it has fared during the market selloff, and read a comparative analysis of SPDR and Vanguard Industrial ETFs.
RELATED FAQS
  1. What is the difference between standard deviation and variance?

    Understand the difference between standard deviation and variance; learn how each is calculated and how these concepts are ... Read Answer >>
  2. What is the difference between standard deviation and Z-score?

    Understand the basics of standard deviation and Z-score, and learn how each is calculated and used in the assessment of market ... Read Answer >>
  3. How do you calculate variance in Excel?

    To calculate statistical variance in Microsoft Excel, use the built-in Excel function VAR. Read Answer >>
  4. What is the best measure of a stock's volatility?

    Understand what metrics are most commonly used to assess a security's volatility compared to its own price history and that ... Read Answer >>
  5. What is a relative standard error?

    Find out how to distinguish between mean, standard deviation, standard error and relative standard error in statistical survey ... Read Answer >>
  6. What is the difference between the standard error of the mean and standard deviation?

    Learn about the difference between the standard error of the mean and the standard deviation and how standard deviation is ... Read Answer >>
Trading Center