Standard deviation

standard deviationssample standard deviationsigmaSDσ&sigma;deviations.d.uncertainty5 Sigma
In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.wikipedia
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Statistics

statisticalstatistical analysisstatistician
In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).

Statistical dispersion

In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

Volatility (finance)

volatilityvolatileprice volatility
The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment.
In finance, volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns.

Robust statistics

robustbreakdown pointrobustness
It is algebraically simpler, though in practice less robust, than the average absolute deviation.
For example, robust methods work well for mixtures of two normal distributions with different standard-deviations; under this model, non-robust methods like a t-test work poorly.

Data set

datasetdatasetsdata
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.
These include the number and types of the attributes or variables, and various statistical measures applicable to them, such as standard deviation and kurtosis.

Square root

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.
It defines an important concept of standard deviation used in probability theory and statistics.

Normal distribution

normally distributednormalGaussian
This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches (7.62 cm) of the mean (67–73 inches (170.18–185.42 cm))one standard deviationand almost all men (about 95%) have a height within 6 inches (15.24 cm) of the mean (64–76 inches (162.56–193.04 cm))two standard deviations.
\sigma is the standard deviation, and

Sigma

Σlunate sigma
In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
the standard deviation of a population or probability distribution in statistics

Bessel's correction

Bessel-corrected
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.

Log-normal distribution

lognormallog-normallognormal distribution
For example, in the case of the log-normal distribution with parameters μ and σ 2, the standard deviation is
Given a log-normally distributed random variable X and two parameters \mu and \sigma that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of X is normally distributed, and we can write X as

Unbiased estimation of standard deviation

anti-biasingsample standard deviationbias
However, unlike in the case of estimating the population mean, for which the sample mean 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.
In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value.

Confidence interval

confidence intervalsconfidence levelconfidence
The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval.
As the machine cannot fill every cup with exactly 250.0 g, the content added to individual cups shows some variation, and is considered a random variable X. This variation is assumed to be normally distributed around the desired average of 250 g, with a standard deviation, σ, of 2.5 g. To determine if the machine is adequately calibrated, a sample of n = 25 cups of liquid is chosen at random and the cups are weighed.

Variance

sample variancepopulation variancevariability
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.
The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, or.

Random variable

random variablesrandom variationrandom
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.
Once the "average value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.
Standard deviation: the square root of the variance, and hence another measure of dispersion.

Statistical significance

statistically significantsignificantsignificantly
In science, many researchers report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from likely genuine effects or associations.
In specific fields such as particle physics and manufacturing, statistical significance is often expressed in multiples of the standard deviation or sigma of a normal distribution, with significance thresholds set at a much stricter level (e.g. 5σ).

Z-test

Z''-teststandardized testingStouffer Z
One can find the standard deviation of an entire population in cases (such as standardized testing) where every member of a population is sampled.
If T is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a Z-test is to estimate the expected value θ of T under the null hypothesis, and then obtain an estimate s of the standard deviation of T.

Mean squared error

mean square errormean-squared errorsquared error loss
This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation.
In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error.

Bias of an estimator

unbiasedunbiased estimatorbias
However, unlike in the case of estimating the population mean, for which the sample mean 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.
Further, mean-unbiasedness is not preserved under non-linear transformations, though median-unbiasedness is (see ); for example, the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.

Statistic

sample statisticempiricalmeasure
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.
Sample variance and sample standard deviation

Kurtosis

excess kurtosisleptokurticplatykurtic
where γ 2 denotes the population excess kurtosis.
where μ 4 is the fourth central moment and σ is the standard deviation.

Estimator

estimatorsestimateestimates
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).
The standard deviation of an estimator of θ (the square root of the variance), or an estimate of the standard deviation of an estimator of θ, is called the standard error of θ.

Square (algebra)

squaresquaredsquares
First, calculate the deviations of each data point from the mean, and square the result of each:
Squaring is used in statistics and probability theory in determining the standard deviation of a set of values, or a random variable.

Margin of error

margins of errorMoE(±err)
For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times.
Given the observed percentage difference p − q (2% or 0.02) and the standard error of the difference calculated above (.03), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0.02 and standard deviation 0.03 is greater than 0.

Prediction interval

interval forecastsPI
See prediction interval.
For example, if one makes the parametric assumption that the underlying distribution is a normal distribution, and has a sample set {X 1, ..., X n }, then confidence intervals and credible intervals may be used to estimate the population mean μ and population standard deviation σ of the underlying population, while prediction intervals may be used to estimate the value of the next sample variable, X n+1.