Standard Deviation Formula and Uses vs. Variance

What Is Standard Deviation?

Standard deviation is a statistical measurement that looks at how far individual points in a dataset are dispersed from the mean of that set. If data points are further from the mean, there is a higher deviation within the data set. It is calculated as the square root of the variance.

How Standard Deviation Works

Standard deviation is a statistical measurement that is often used in finance, particularly in investing. When applied to the annual rate of return of an investment, it can provide information on that investment's historical volatility. This means that it shows how much the price of that investment has fluctuated over time.

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, meaning that its price goes up and down frequently. The standard deviation of a stable blue-chip stock, on the other hand, is usually rather low, meaning that its price is usually stable.

Standard deviation can also be used to predict performance trends. In investing, for example, an index fund is designed to replicate a benchmark index. This means that the fund will have a low standard deviation from the value of the benchmark.

On the other hand, aggressive growth funds often have a high standard deviation from relative stock indices. This is because their portfolio managers make aggressive bets to generate higher-than-average returns. This higher standard deviation correlates with the level of risk investors can expect from that index.

Standard deviation is one of the key fundamental risk measures that analysts, portfolio managers, and 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 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:

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

Key Properties of Standard Deviation

One key property of standard deviation is additivity. This means that the standard deviation of a sum of random variables. This means that analysts or researchers using standard deviation are comparing many data points, rather than drawing conclusions based on only analyzing single points of data, which leads to a higher degree of accuracy.

Another property of standard deviation is scale invariance. This is particularly useful in comparing the variability of datasets with different units of measurement. For example, if one dataset is measured in inches and another in centimeters, their standard deviations can still be compared directly without needing to convert units.

Last, standard deviation has properties of symmetry and non-negativity. This means a standard deviation is always positive and symmetrically distributed around the mean. This symmetry property implies that deviations above the mean are balanced by deviations below the mean, resulting in a total balance of the entire data set. The property of always being positive means a standard deviation has a higher degree of comparability when looking at standard deviations across data sets.

Standard Deviation vs. Variance

Variance and standard deviation are related statistics. 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.

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.

How Standard Deviation Is Used in Business

Standard deviation isn't only used in investing. Business analysts or companies can use standard deviation in a variety of ways to assess risk, make predictions, and manage company operations.

Risk Management

Standard deviation is widely used in business for risk management. It helps businesses quantify and manage various types of risks. By calculating the standard deviation of certain outcomes, businesses can assess the volatility or uncertainty associated with how they operates. For example, a company can use standard deviation to measure the risk of different products being returned.

Financial Analysis

In finance and accounting, standard deviation is used to analyze financial data and assess the variability of financial performance metrics. For example, standard deviation is employed to measure the volatility of investment returns. This can used to determine risk-return tradeoffs and the strategy of how a company wants to deploy capital.

Forecasting

Standard deviation is used in sales forecasting to assess the variability of sales data and predict future sales trends. Standard deviation helps businesses identify seasonality, trends, and patterns in sales data that allow them to plan for cash needs in the near future.

Quality Control

In manufacturing and operations management, standard deviation is used to monitor and improve product quality. Standard deviation is also used in quality control processes such as Six Sigma methodologies to measure process capability, reduce defects, and optimize manufacturing processes for improved quality and customer satisfaction.

Project Management

Standard deviation is used in project management to assess project performance and manage risks. For example, standard deviation can be used related to critical path analysis and earned value. It can used to gauge variances, track progress, and quantify risk related to a critical path or earned value not being achieved.

Strengths and Limitations of Standard Deviation

Like any statistical measurement for analyzing data, standard deviation has both strengths and limitations that should be considered before it is used.

Strengths

• Commonly used: 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 used by a variety of professions, from investors to actuaries.
• Includes all data points: 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.
• Can combine datasets: The standard deviation of two data sets can be combined using a specific combined standard deviation formula. There are no similar formulas for other dispersion observation measurements in statistics.
• Further computational uses: Unlike other means of observation, the standard deviation can be used in further algebraic computations, meaning there's some versatility to standard deviation.

Limitations

• Doesn't measure dispersion: 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.
• Impact of outliers: 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.
• Difficult to calculate manually: As opposed to other measurements of dispersion such as range (the highest value minus 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.

Examples of Standard Deviation

If you have the data points 5, 7, 3, and 7 and want to find the standard deviation, start by adding them together:

5 + 7 + 3 + 7 = 22

Find the mean of the dataset by dividing the total by the number of data points (in this case, 4).

22 / 4 = 5.5

This gives you x̄ = 5.5 and N = 4.

To find the variance, subtract the mean value from each data point, then square each of those values:

5 - 5.5 = -0.5 x -0.5 = 0.25

7 - 5.5 = 1.5 x 1.5 = 2.25

3 - 5.5 = -2.5 x -2.5 = 6.25

7 - 5.5 = 1.5 x 1.5 = 2.25

Add the square values, then divide the result by N-1 to give the variance.

(0.25 + 2.25 + 6.25 + 2.25) / (4-1) = 3.67

Take the square root of the 3.67 to find the standard deviation, which is approximately 1.915.

Or consider shares of Apple (AAPL) over five years. Historical returns for Apple’s stock were 88.97% for 2019, 82.31% for 2020, 34.65% for 2021, -26.41% for 2022 and 28.32% in April 2023. The average return over the five years was thus 41.57%.

The value of each year's return minus the mean were then 47.40%, 40.74%, -6.92%, -67.98%, and -15.57%, respectively. All those values are then squared to yield 22.47%, 16.60%, 0.48%, 46.21%, and 2.42%. The sum of these values is 0.882. Divide that value by 4 (N minus 1) to get the variance (0.882/4) = 0.220.

The square root of the variance is taken to obtain the standard deviation of 0.4690, or 46.90%.

The Bottom Line

Standard deviation is a way to assess risk, especially in business and investing. It uses the distance of points in a dataset from the mean of that dataset to find how dispersed the set is, and thus, how volatile it tends to be over time.

Investors can use standard deviation to determine how stable or predictable an investment is likely to be. Businesses use standard deviation or assess risk, manage operations, and plan cash flows. Like any other statistical measurement, standard deviation has strengths and limitations, which should be taken into account when it is used.