TheInfoList

OR:  In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the ''arithm ...
(also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case
Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
σ (sigma), for the population standard deviation, or the Latin letter '' s'', for the sample standard deviation. The standard deviation of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, sample,
statistical population In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypoth ...
, data set, or probability distribution is the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of its variance. It is
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a population or sample and the
standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...
of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the
margin of error The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a census of the e ...
of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population. In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant", a safeguard against spurious conclusion that is really due to random sampling error. When only a sample of data from a population is available, the term ''standard deviation of the sample'' or ''sample standard deviation'' can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the ''population standard deviation'' (the standard deviation of the entire population).

# Basic examples

## Population standard deviation of grades of eight students

Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of the average of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (that is, a
statistical population In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypoth ...
) are the following eight values: : $2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9.$ These eight data points have the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the ''arithm ...
(average) of 5: : $\mu = \frac = \frac = 5.$ First, calculate the deviations of each data point from the mean, and
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
the result of each: : $\begin \left(2 - 5\right)^2 = \left(-3\right)^2 = 9 && \left(5 - 5\right)^2 = 0^2 = 0 \\ \left(4 - 5\right)^2 = \left(-1\right)^2 = 1 && \left(5 - 5\right)^2 = 0^2 = 0 \\ \left(4 - 5\right)^2 = \left(-1\right)^2 = 1 && \left(7 - 5\right)^2 = 2^2 = 4 \\ \left(4 - 5\right)^2 = \left(-1\right)^2 = 1 && \left(9 - 5\right)^2 = 4^2 = 16. \\ \end$ The variance is the mean of these values: :$\sigma^2 = \frac = \frac = 4.$ and the ''population'' standard deviation is equal to the square root of the variance: : $\sigma = \sqrt = 2.$ This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by instead of in the denominator of the last formula, and the result is $s = \sqrt \approx 2.1.$ In that case, the result of the original formula would be called the ''sample'' standard deviation and denoted by ''s'' instead of $\sigma.$ Dividing by ''n'' − 1 rather than by ''n'' gives an unbiased estimate of the variance of the larger parent population. This is known as ''
Bessel's correction In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
''. Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by ''n'' would underestimate the variability.

## Standard deviation of average height for adult men

If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
is about 70 inches, with a standard deviation of around 3 inches. This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches tall. If the standard deviation were 20 inches, then men would have much more variable heights, with a typical range of about 50–90 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule, or the ''empirical rule,'' for more information).

# Definition of population values

Let ''μ'' be the expected value (the average) of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' with density ''f''(''x''): The standard deviation ''σ'' of ''X'' is defined as $\sigma \equiv \sqrt = \sqrt,$ which can be shown to equal $\sqrt.$ Using words, the standard deviation is the square root of the variance of ''X''. The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuari ...
with parameter $\alpha \in \left(1,2\right]$ has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
has neither a mean nor a standard deviation.

## Discrete random variable

In the case where ''X'' takes random values from a finite data set ''x''1, ''x''2, …, ''xN'', with each value having the same probability, the standard deviation is $\sigma = \sqrt,\text \mu = \frac (x_1 + \cdots + x_N),$ or, by using summation notation, $\sigma = \sqrt,\text \mu = \frac \sum_^N x_i.$ If, instead of having equal probabilities, the values have different probabilities, let ''x''1 have probability ''p''1, ''x''2 have probability ''p''2, …, ''x''''N'' have probability ''p''''N''. In this case, the standard deviation will be $\sigma = \sqrt,\text \mu = \sum_^N p_i x_i.$

## Continuous random variable

The standard deviation of a continuous real-valued random variable ''X'' with
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''p''(''x'') is $\sigma = \sqrt,\text \mu = \int_\mathbf x\, p(x)\, \mathrm dx,$ and where the integrals are definite integrals taken for ''x'' ranging over the set of possible values of the random variable ''X''. In the case of a parametric family of distributions, the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters ''μ'' and ''σ''2, the standard deviation is $\sqrt.$

# Estimation

One can find the standard deviation of an entire population in cases (such as standardized testing) where every member of a population is sampled. In cases where that cannot be done, the standard deviation ''σ'' is estimated by examining a random sample taken from the population and computing a statistic of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an estimator, and the estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by ''s'' (possibly with modifiers). Unlike in the case of estimating the population mean, for which the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger po ...
is a simple estimator with many desirable properties ( unbiased, efficient, maximum likelihood), there is no single estimator for the standard deviation with all these properties, and unbiased estimation of standard deviation is a very technically involved problem. Most often, the standard deviation is estimated using the '' corrected sample standard deviation'' (using ''N'' − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using ''N'') yields lower mean squared error, while using ''N'' − 1.5 (for the normal distribution) almost completely eliminates bias.

## Uncorrected sample standard deviation

The formula for the ''population'' standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by ''s''''N'', is known as the ''uncorrected sample standard deviation'', or sometimes the ''standard deviation of the sample'' (considered as the entire population), and is defined as follows: $s_N = \sqrt,$ where $\$ are the observed values of the sample items, and $\bar$ is the mean value of these observations, while the denominator ''N'' stands for the size of the sample: this is the square root of the sample variance, which is the average of the
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average val ...
about the sample mean. This is a consistent estimator (it converges in probability to the population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/''N'', and thus is most significant for small or moderate sample sizes; for $N > 75$ the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
than the corrected sample standard deviation.

## Corrected sample standard deviation

If the ''biased sample variance'' (the second
central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is :$s_N = \sqrt.$ Here taking the square root introduces further downward bias, by
Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question. An unbiased estimator for the ''variance'' is given by applying
Bessel's correction In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
, using ''N'' − 1 instead of ''N'' to yield the ''unbiased sample variance,'' denoted ''s''2: :$s^2 = \frac \sum_^N \left\left(x_i - \bar\right\right)^2.$ This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement. ''N'' − 1 corresponds to the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
in the vector of deviations from the mean, $\textstyle\left(x_1 - \bar,\; \dots,\; x_n - \bar\right).$ Taking square roots reintroduces bias (because the square root is a nonlinear function which does not commute with the expectation, i.e. often ), yielding the ''corrected sample standard deviation,'' denoted by ''s:'' :$s = \sqrt.$ As explained above, while ''s''2 is an unbiased estimator for the population variance, ''s'' is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples (''N'' less than 10). As sample size increases, the amount of bias decreases. We obtain more information and the difference between $\frac$ and $\frac$ becomes smaller.

## Unbiased sample standard deviation

For unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, ''s'' is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by ''s''/''c''4, where the correction factor (which depends on ''N'') is given in terms of the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, and equals: :$c_4\left(N\right)\,=\,\sqrt\,\,\,\frac.$ This arises because the sampling distribution of the sample standard deviation follows a (scaled)
chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard nor ...
, and the correction factor is the mean of the chi distribution. An approximation can be given by replacing ''N'' − 1 with ''N'' − 1.5, yielding: : $\hat\sigma = \sqrt,$ The error in this approximation decays quadratically (as 1/''N''2), and it is suited for all but the smallest samples or highest precision: for ''N'' = 3 the bias is equal to 1.3%, and for ''N'' = 9 the bias is already less than 0.1%. A more accurate approximation is to replace $N-1.5$ above with $N-1.5+1/\left(8\left(N-1\right)\right)$. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: : $\hat\sigma = \sqrt,$ where ''γ''2 denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.

## Confidence interval of a sampled standard deviation

The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the
confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
or CI. To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of ''N'' = 2 has only 1 degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows: :$\Pr\left\left(q_\frac < k \frac < q_\right\right) = 1 - \alpha,$ where $q_p$ is the ''p''-th quantile of the chi-square distribution with ''k'' degrees of freedom, and $1 - \alpha$ is the confidence level. This is equivalent to the following: :$\Pr\left\left(k\frac < \sigma^2 < k\frac\right\right) = 1 - \alpha.$ With ''k'' = 1, $q_ = 0.000982$ and $q_ = 5.024$. The reciprocals of the square roots of these two numbers give us the factors 0.45 and 31.9 given above. A larger population of ''N'' = 10 has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69 × SD to 1.83 × SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a sample population N=100, this is down to 0.88 × SD to 1.16 × SD. To be more certain that the sampled SD is close to the actual SD we need to sample a large number of points. These same formulae can be used to obtain confidence intervals on the variance of residuals from a least squares fit under standard normal theory, where ''k'' is now the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
for error.

## Bounds on standard deviation

For a set of ''N'' > 4 data spanning a range of values ''R'', an upper bound on the standard deviation ''s'' is given by ''s = 0.6R''. An estimate of the standard deviation for ''N'' > 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values ''R'' represents four standard deviations so that ''s ≈ R/4''. This so-called range rule is useful in
sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
estimation, as the range of possible values is easier to estimate than the standard deviation. Other divisors ''K(N)'' of the range such that ''s ≈ R/K(N)'' are available for other values of ''N'' and for non-normal distributions.

# Identities and mathematical properties

The standard deviation is invariant under changes in
location In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
, and scales directly with the scale of the random variable. Thus, for a constant ''c'' and random variables ''X'' and ''Y'': : $\begin \sigma\left(c\right) &= 0 \\ \sigma\left(X + c\right) &= \sigma\left(X\right), \\ \sigma\left(cX\right) &= , c, \sigma\left(X\right). \end$ The standard deviation of the sum of two random variables can be related to their individual standard deviations and the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
between them: : $\sigma\left(X + Y\right) = \sqrt. \,$ where $\textstyle\operatorname \,=\, \sigma^2$ and $\textstyle\operatorname$ stand for variance and
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, respectively. The calculation of the sum of squared deviations can be related to moments calculated directly from the data. In the following formula, the letter E is interpreted to mean expected value, i.e., mean. :$\sigma\left(X\right) = \sqrt = \sqrt.$ The sample standard deviation can be computed as: :$s\left(X\right) = \sqrt \sqrt.$ For a finite population with equal probabilities at all points, we have :$\sqrt = \sqrt = \sqrt,$ which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value. See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.

# Interpretation and application A large standard deviation indicates that the data points can spread 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 , and 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. These standard deviations have the same units as the data points themselves. If, for instance, the data set represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters. Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
s 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. See
prediction interval In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
. While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the mean absolute deviation, which might be considered a more direct measure of average distance, compared to the root mean square distance inherent in the standard deviation.

## Application examples

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).

### Experiment, industrial and hypothesis testing

Standard deviation is often used to compare real-world data against a model to test the model. For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (99.9% or more). If it falls outside the range then the production process may need to be corrected. Statistical tests such as these are particularly important when the testing is relatively expensive. For example, if the product needs to be opened and drained and weighed, or if the product was otherwise used up by the test. In experimental science, a theoretical model of reality is used.
Particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and ...
conventionally uses a standard of "5 sigma" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to assert that a particle consistent with the
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
had been discovered in two independent experiments at
CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
, also leading to the declaration of the first observation of gravitational waves.

### Weather

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.

### Finance

In finance, standard deviation is often used as a measure of the
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets (actively managed mutual funds, index mutual funds, or ETFs). 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 (known as mean-variance optimization). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the risk premium. In other words, investors should expect a higher return on an investment when that investment carries a higher level of risk or uncertainty. 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. For example, assume an investor had to choose between two stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20
percentage point A percentage point or percent point is the unit for the arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points, but a 10-percent increase in the quantity being me ...
s (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return). Stock B is likely to fall short of the initial investment (but also to exceed the initial investment) more often than Stock A under the same circumstances, and is estimated to return only two percent more on average. In this example, Stock A is expected to earn about 10 percent, plus or minus 20 pp (a range of 30 percent to −10 percent), about two-thirds of the future year returns. When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70 percent to −50 percent, which includes outcomes for three standard deviations from the average return (about 99.7 percent of probable returns). Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average gives the overall variance of the return of the asset. The larger the variance, the greater risk the security carries. Finding the square root of this variance will give the standard deviation of the investment tool in question. Population standard deviation is used to set the width of Bollinger Bands, a
technical analysis In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sam ...
tool. For example, the upper Bollinger Band is given as $\textstyle\bar + n\sigma_x.$ The most commonly used value for ''n'' is 2; there is about a five percent chance of going outside, assuming a normal distribution of returns. Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.

## Geometric interpretation

To gain some geometric insights and clarification, we will start with a population of three values, ''x''1, ''x''2, ''x''3. This defines a point ''P'' = (''x''1, ''x''2, ''x''3) in R3. Consider the line ''L'' = . This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and ''P'' would lie on ''L''. So it is not unreasonable to assume that the standard deviation is related to the ''distance'' of ''P'' to ''L''. That is indeed the case. To move orthogonally from ''L'' to the point ''P'', one begins at the point: :$M = \left\left(\bar, \bar, \bar\right\right)$ whose coordinates are the mean of the values we started out with. $M$ is on $L$ therefore $M = \left(\ell,\ell,\ell\right)$ for some $\ell \in \mathbb$. The line $L$ is to be orthogonal to the vector from $M$ to $P$. Therefore: : A little algebra shows that the distance between ''P'' and ''M'' (which is the same as the orthogonal distance between ''P'' and the line ''L'') $\sqrt$ is equal to the standard deviation of the vector (''x''1, ''x''2, ''x''3), multiplied by the square root of the number of dimensions of the vector (3 in this case).

## Chebyshev's inequality

An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.

## Rules for normally distributed data The
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of :$f\left\left(x, \mu, \sigma^2\right\right) = \frac e^$ where ''μ'' is the expected value of the random variables, ''σ'' equals their distribution's standard deviation divided by ''n''1/2, and ''n'' is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the
normalizing constant The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ...
. If a data distribution is approximately normal, then the proportion of data values within ''z'' standard deviations of the mean is defined by: : $\text = \operatorname\left\left(\frac\right\right)$ where $\textstyle\operatorname$ is the error function. The proportion that is less than or equal to a number, ''x'', is given by the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
: :  # Relationship between standard deviation and mean

The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of
statistical dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquart ...
if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose ''x''1, ..., ''x''''n'' are real numbers and define the function: :$\sigma\left(r\right) = \sqrt.$ Using
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
or by completing the square, it is possible to show that ''σ''(''r'') has a unique minimum at the mean: :$r = \bar.\,$ Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. It is a dimensionless number.

## Standard deviation of the mean

Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: :$\sigma_\text = \frac\sigma$ where ''N'' is the number of observations in the sample used to estimate the mean. This can easily be proven with (see basic properties of the variance): :$\begin \operatorname\left(X\right) &\equiv \sigma^2_X\\ \operatorname\left(X_1 + X_2\right) &\equiv \operatorname\left(X_1\right) + \operatorname\left(X_2\right)\\ \end$ (Statistical independence is assumed.) :$\operatorname\left(cX_1\right) \equiv c^2\, \operatorname\left(X_1\right)$ hence :$\begin \operatorname\left(\text\right) &= \operatorname\left\left(\frac\sum_^N X_i\right\right) = \frac \operatorname\left\left(\sum_^N X_i\right\right) \\ &= \frac \sum_^N \operatorname\left(X_i\right) = \frac \operatorname\left(X\right) = \frac \operatorname\left(X\right). \end$ Resulting in: :$\sigma_\text = \frac.$ In order to estimate the standard deviation of the mean $\sigma_\text$ it is necessary to know the standard deviation of the entire population $\sigma$ beforehand. However, in most applications this parameter is unknown. For example, if a series of 10 measurements of a previously unknown quantity is performed in a laboratory, it is possible to calculate the resulting sample mean and sample standard deviation, but it is impossible to calculate the standard deviation of the mean. However, one can estimate the standard deviation of the entire population from the sample, and thus obtain an estimate for the standard error of the mean.

# Rapid calculation methods

The following two formulas can represent a running (repeatedly updated) standard deviation. A set of two power sums ''s''1 and ''s''2 are computed over a set of ''N'' values of ''x'', denoted as ''x''1, ..., ''x''''N'': :$s_j = \sum_^N.$ Given the results of these running summations, the values ''N'', ''s''1, ''s''2 can be used at any time to compute the ''current'' value of the running standard deviation: :$\sigma = \frac$ Where N, as mentioned above, is the size of the set of values (or can also be regarded as ''s''0). Similarly for sample standard deviation, :$s = \sqrt.$ In a computer implementation, as the two ''s''''j'' sums become large, we need to consider
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
,
arithmetic overflow Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors. This is a "one pass" algorithm for calculating variance of ''n'' samples without the need to store prior data during the calculation. Applying this method to a time series will result in successive values of standard deviation corresponding to ''n'' data points as ''n'' grows larger with each new sample, rather than a constant-width sliding window calculation. For ''k'' = 1, ..., ''n'': :$\begin A_0 &= 0\\ A_k &= A_ + \frac \end$ where A is the mean value. :$\begin Q_0 &= 0 \\ Q_k &= Q_ + \frac \left\left(x_k - A_\right\right)^2 = Q_ + \left\left(x_k - A_\right\right)\left\left(x_k - A_k\right\right) \end$ Note: $Q_1 = 0$ since $k-1 = 0$ or $x_1 = A_1$ Sample variance: :$s^2_n = \frac$ Population variance: :$\sigma^2_n = \frac$

## Weighted calculation

When the values ''xi'' are weighted with unequal weights ''wi'', the power sums ''s''0, ''s''1, ''s''2 are each computed as: :$s_j = \sum_^N w_k x_k^j.\,$ And the standard deviation equations remain unchanged. ''s''0 is now the sum of the weights and not the number of samples ''N''. The incremental method with reduced rounding errors can also be applied, with some additional complexity. A running sum of weights must be computed for each ''k'' from 1 to ''n'': :$\begin W_0 &= 0 \\ W_k &= W_ + w_k \end$ and places where 1/''n'' is used above must be replaced by ''wi''/''Wn'': :$\begin A_0 &= 0 \\ A_k &= A_ + \frac\left\left(x_k - A_\right\right) \\ Q_0 &= 0 \\ Q_k &= Q_ + \frac\left\left(x_k -A_\right\right)^2 = Q_ + w_k\left\left(x_k-A_\right\right)\left\left(x_k - A_k\right\right) \end$ In the final division, :$\sigma^2_n = \frac\,$ and :$s^2_n = \frac,$ or :$s^2_n = \frac \sigma^2_n,$ where ''n'' is the total number of elements, and ''n is the number of elements with non-zero weights. The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.

# History

The term ''standard deviation'' was first used in writing by
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
in 1894, following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea: for example,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
used ''mean error''.

# Higher dimensions In two dimensions, the standard deviation can be illustrated with the standard deviation ellipse (see '' Multivariate normal distribution § Geometric interpretation'').

* 68–95–99.7 rule *
Accuracy and precision Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
*
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...
An inequality on location and scale parameters * Coefficient of variation *
Cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
*
Deviation (statistics) In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is posi ...
* Distance correlation Distance standard deviation * Error bar * Geometric standard deviation *
Mahalanobis distance The Mahalanobis distance is a measure of the distance between a point ''P'' and a distribution ''D'', introduced by P. C. Mahalanobis in 1936. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls base ...
generalizing number of standard deviations to the mean *
Mean absolute error In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of ''Y'' versus ''X'' include comparisons of predicted versus observed, subsequent time versus initial time, and ...
* Pooled variance * Propagation of uncertainty *
Percentile In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage falls ...
*
Raw data Raw data, also known as primary data, are ''data'' (e.g., numbers, instrument readings, figures, etc.) collected from a source. In the context of examinations, the raw data might be described as a raw score (after test scores). If a scientis ...
* Robust standard deviation *
Root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
*
Sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
* Samuelson's inequality * Six Sigma *
Standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...
*
Standard score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mea ...
* Yamartino method for calculating standard deviation of wind direction