Standard Deviation Calculation

Standard Deviation

Investopedia / Alex Dos Diaz

What Is Standard Deviation?

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. The standard deviation is calculated as the square root of variance by determining each data point's deviation 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.

Key Takeaways:

  • Standard deviation measures the dispersion of a dataset relative to its mean.
  • It is calculated as the square root of the variance.
  • Standard deviation, in finance, is often used as a measure of a relative riskiness of an asset.
  • A volatile stock has a high standard deviation, while the deviation of a stable blue-chip stock is usually rather low.
  • As a downside, the standard deviation calculates all uncertainty as risk, even when it’s in the investor's favor—such as above-average returns.
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Standard Deviation

Understanding 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 that investment's historical volatility.

The greater the standard deviation of securities, 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.

Standard Deviation Formula

Standard deviation is calculated by taking the square root of a value derived from comparing data points to a collective mean of a population. The formula is:

Standard Deviation = i = 1 n ( x i x ) 2 n 1 where: x i = Value of the  i t h  point in the data set x = The mean value of the data set n = The number of data points in the data set \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=n1i=1n(xix)2where:xi=Value of the ith point in the data setx=The mean value of the data setn=The number of data points in the data set

Calculating Standard Deviation

Standard deviation is calculated as follows:

  1. Calculate the mean of all data points. The mean is calculated by adding all the data points and dividing them by the number of data points.
  2. Calculate the variance for each data point. The variance for each data point is calculated by subtracting the mean from the value of the data point.
  3. Square the variance of each data point (from Step 2).
  4. Sum of squared variance values (from Step 3).
  5. Divide the sum of squared variance values (from Step 4) by the number of data points in the data set less 1.
  6. Take the square root of the quotient (from Step 5).

Using Standard Deviation

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 is likely 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 and the investor's willingness to assume risk. When dealing with the amount of deviation in their portfolios, investors should consider their 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.

Standard Deviation vs. Variance

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. However, this is more difficult to grasp than the standard deviation 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. Larger variances cause more data points to fall outside the standard deviation. Smaller variances result in more data that is close to average.

The standard deviation is graphically depicted as a bell curve's width around the mean of a data set. The wider the curve's width, the larger a data set's standard deviation from the mean.

Strengths of Standard Deviation

Standard deviation is a commonly used measure of dispersion. Many analysts are probably more familiar with standard deviation than compared to other statistical calculations of data deviation. For this reason, the standard deviation is often used in a variety of situations from investing to actuaries.

Standard deviation is all-inclusive of observations. Each data point is included in the analysis. Other measurements of deviation such as range only measure the most dispersed points without consideration for the points in between. Therefore, standard deviation is often considered a more robust, accurate measurement compared to other observations.

The standard deviation of two data sets can be combined using a specific combined standard deviation formula. There is no similar formulas for other dispersion observation measurements in statistics. In addition, the standard deviation can be used in further algebraic computations unlike other means of observation.

Limitations of Standard Deviation

There are some downsides to consider when using standard deviation. The standard deviation does not actually measure how far a data point is from the mean. Instead, it compares the square of the differences, a subtle but notable difference from actual dispersion from the mean.

Outliers have a heavier impact on standard deviation. This is especially true considering the difference from the mean is squared, resulting in an even larger quantity compared to other data points. Therefore, be mindful that standard observation naturally gives more weight to extreme values.

Last, standard deviation can be difficult to manually calculate. As opposed to other measurements of dispersion such as range (the highest value less the lowest value), standard deviation requires several cumbersome steps and is more likely to incur computational errors compared to easier measurements. This hurdle can be circumnavigated through the use of a Bloomberg terminal.

Consider leveraging Excel when calculating standard deviation. After entering your data, use the STDEV.S formula if your data set is numeric or the STDEVA when you want to include text or logical values. There are also several specific formulas to calculate the standard deviation for an entire population.

Example of Standard Deviation

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 mean's value 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, giving a total of 11, which is then divided by the value of N minus 1, which is 3, resulting in a variance of approximately 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 (AAPL) for a period of five years. Historical returns for Apple’s stock were 12.49% for 2016, 48.45% for 2017, -5.39% for 2018, 88.98% for 2019 and, as of September, 60.91% for 2020. The average return over the five years was thus 41.09%.

The value of each year's return less the mean were then -28.6%, 7.36% -46.48%, 47.89%, and 19.82%, respectively. All those values are then squared to yield 8.2%, 0.54%, 21.6%, 22.93%, and 3.93%. The sum of these values is 0.572. Divide that value by 4 (N minus 1) to get the variance (0.572/4) = 0.143. The square root of the variance is taken to obtain the standard deviation of 0.3781, or 37.81%.

What Does a High Standard Deviation Mean?

A large standard deviation indicates that there is a lot of variance in the observed data around the mean. This indicates that the data observed is quite spread out. A small or low standard deviation would indicate instead that much of the data observed is clustered tightly around the mean.

What Does Standard Deviation Tell You?

Standard deviation describes how dispersed a set of data is. It compares each data point to the mean of all data points, and standard deviation returns a calculated value that describes whether the data points are in close proximity or whether they are spread out. In a normal distribution, standard deviation tells you how far values are from the mean.

How Do You Find the Standard Deviation Quickly?

If you look at the distribution of some observed data visually, you can see if the shape is relatively skinny vs. fat. Fatter distributions have bigger standard deviations. Alternatively, Excel has built in standard deviation functions depending on the data set.

How Do You Calculate Standard Deviation?

Standard deviation is calculated as the square root of the variance. Alternatively, it is calculated by finding the mean of a data set, finding the difference of each data point to the mean, squaring the differences, adding them together, dividing by the number of points in the data set less 1, and finding the square root.

Why Is Standard Deviation Important?

Standard deviation is important because it can help users assess risk. Consider an investment option with an average annual return of 10% per year. However, this average was derived from the past three year returns of 50%, -15%, and -5%. By calculating the standard deviation and understanding your low likelihood of actually averaging 10% in any single given year, you're better armed to make informed decisions and recognizing underlying risk.

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