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Interpreting the Standard Deviation
The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the mean.
Learning Objective

Derive standard deviation to measure the uncertainty in daily life examples
Key Points
 A large standard deviation indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean.
 When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance.
 In finance, standard deviation is often used as a measure of the risk associated with pricefluctuations of a given asset (stocks, bonds, property, etc. ), or the risk of a portfolio of assets.
Terms

disparity
the state of being unequal; difference

standard deviation
a measure of how spread out data values are around the mean, defined as the square root of the variance
Example
 In finance, standard deviation is often used as a measure of the risk associated with pricefluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets. Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns.
Full Text
Interpretation of the Standard Deviation
A large standard deviation, which is the square root of the variance, indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean. For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7.
Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance. If the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified.
Application of the Standard Deviation
The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).
Climate
As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.
Sports
Another way of seeing it is to consider sports teams. In any set of categories, there will be teams that rate highly at some things and poorly at others. Chances are, the teams that lead in the standings will not show such disparity but will perform well in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent they will tend to be. Teams with a higher standard deviation, however, will be more unpredictable.
Comparison of Standard Deviations
Example of two samples with the same mean and different standard deviations. The red sample has a mean of 100 and a SD of 10; the blue sample has a mean of 100 and a SD of 50. Each sample has 1,000 values drawn at random from a Gaussian distribution with the specified parameters.
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Key Term Reference
 Precision
 Appears in these related concepts: Accuracy, Precision, and Error, Principles of Writing in the Sciences, and Accuracy vs. Precision
 average
 Appears in these related concepts: Outliers, Mean: The Average, and Average Value of a Function
 datum
 Appears in these related concepts: Inferential Statistics, Misleading Graphs, and Change of Scale
 deviation
 Appears in these related concepts: Applications of Statistics, Graphs of Qualitative Data, and Variance
 factor
 Appears in these related concepts: The Perceptual Process, Finding Factors of Polynomials, and Solving Quadratic Equations by Factoring
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 population
 Appears in these related concepts: Organismal Ecology and Population Ecology, What constitutes a biome?, and Basic Inferential Statistics
 range
 Appears in these related concepts: Range, The Derivative as a Function, and Visualizing Domain and Range
 variance
 Appears in these related concepts: Testing a Single Variance, Variance Estimates, and Variance
Sources
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Cite This Source
Source: Boundless. “Interpreting the Standard Deviation.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 13 Feb. 2016 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/measuresofvariation6/describingvariability26/interpretingthestandarddeviation1344425/