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Encyclopedia > Standard deviation

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of statistical dispersion of its values. The standard deviation is usually denoted with the letter σ (lower case sigma). It is defined as the square root of the variance. Probability is the likelihood or chance that something is the case or will happen. ... This article is about the field of statistics. ... A probability distribution describes the values and probabilities that a random event can take place. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In mathematics, a multiset (or bag) is a generalization of a set. ... In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ... Sigma (upper case Σ, lower case σ, alternative ς) is the 18th letter of the Greek alphabet. ... In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... This article is about mathematics. ...


To understand standard deviation, keep in mind that variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared. Standard deviation, being the square root of that quantity, therefore measures the spread of data about the mean, measured in the same units as the data.


Stated more formally, the standard deviation is the root mean square (RMS) deviation of values from their arithmetic mean. In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...


For example, in the population {4, 8}, the mean is 6 and the deviations from mean are {−2, 2}. Those deviations squared are {4, 4} the average of which (the variance) is 4. Therefore, the standard deviation is 2. In this case 100% of the values in the population are at one standard deviation from the mean.


Discovered by Galton in the late 1860s[1], the standard deviation remains the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If many data points are close to the mean, then the standard deviation is small; if many data points are far from the mean, then the standard deviation is large. If all the data values are equal, then the standard deviation is zero. Francis Galton Sir Francis Galton FRS (February 16, 1822 - January 17, 1911) was an English explorer, statistician, anthropologist, creator of modern eugenics (he coined the term), and investigator of the human mind. ... In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ... A data set (or dataset) is a collection of data, usually presented in tabular form. ...


For a population, the standard deviation can be estimated by a modified standard deviation (s) of a sample. The formulae are given below. A sample is that part of a population which is actually observed. ...

Given a random variable (in blue), the standard deviation σ is a measure of the spread of the values of the random variable away from its mean μ.
Given a random variable (in blue), the standard deviation σ is a measure of the spread of the values of the random variable away from its mean μ.

Contents

Image File history File links Standard_deviation. ... Image File history File links Standard_deviation. ...

Definition and calculation

A simple example

Suppose we wished to find the standard deviation of the set of the numbers 4 and 8.


Step 1: find the arithmetic mean (or average) of 4 and 8,

(4 + 8) / 2 = 6.

Step 2: find the deviation of each number from the mean,

4 − 6 = − 2
8 − 6 = 2.

Step 3: square each of the deviations (amplifying larger deviations and making negative values positive),

( − 2)2 = 4
22 = 4.

Step 4: sum the obtained squares (as a first step to obtaining an average),

4 + 4 = 8.

Step 5: divide the sum by the number of values, which here is 2 (giving an average),

8 / 2 = 4.

Step 6: take the non-negative square root of the quotient (converting squared units back to regular units),

sqrt{4}=2.

So, the standard deviation of the set is 2.


Standard deviation of a random variable

The standard deviation of a random variable X is defined as: In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

begin{array}{lcl} sigma & = &sqrt{operatorname{E}((X - operatorname{E}(X))^2)} = sqrt{operatorname{E}(X^2) - (operatorname{E}(X))^2}  & = & sqrt{operatorname{Var}(X)} end{array}

where E(X) is the expected value of X (another word for mean, a kind of average), and Var(X) is the variance of X. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about mathematical mean. ... Averages redirects here. ... This article is about mathematics. ...


Not all random variables have a standard deviation, since these expected values need not exist. For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined because its E(X) is undefined. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...


If the random variable X takes on the values scriptstyle x_1,dots,x_N (which are real numbers) with equal probability, then its standard deviation can be computed as follows. First, the mean of X, overline{x}, is defined as a summation: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about mathematical mean. ... Sum redirects here. ...

overline{x} = frac{1}{N}sum_{i=1}^N x_i = frac{x_1+x_2+cdots+x_N}{N}

where N is the number of samples taken. Next, the standard deviation simplifies to

sigma = sqrt{frac{1}{N} sum_{i=1}^N (x_i - overline{x})^2}.

In other words, the standard deviation of a discrete uniform random variable X can be calculated as follows:

  1. For each value xi calculate the difference scriptstyle x_i - overline{x} between xi and the average value scriptstyleoverline{x}.
  2. Calculate the squares of these differences.
  3. Find the average of the squared differences. This quantity is the variance σ2.
  4. Take the square root of the variance.

The above expression can be replaced with This article is about mathematics. ...

sigma = sqrt{frac{1}{N} left(sum_{i=1}^N x_i^2 - Noverline{x}^2right)}.

Equality of these two expressions can be shown by a bit of algebra:

begin{align} sum_{i=1}^N (x_i - overline{x})^2 & = {} sum_{i=1}^N (x_i^2 - 2 x_ioverline{x} + overline{x}^2)  & {} = left(sum_{i=1}^N x_i^2right) - left(2 overline{x} sum_{i=1}^N x_iright) + Noverline{x}^2  & {} = left(sum_{i=1}^N x_i^2right) - 2 overline{x} (Noverline{x}) + Noverline{x}^2  & {} = left(sum_{i=1}^N x_i^2right) - 2Noverline{x}^2 + Noverline{x}^2  & {} = left(sum_{i=1}^N x_i^2right) - Noverline{x}^2. end{align}

Estimating population standard deviation from sample standard deviation

In the real world, finding the standard deviation of an entire population is unrealistic except in certain cases, such as standardized testing, where every member of a population is sampled. In most cases, the standard deviation is estimated by examining a random sample taken from the population. The most common measure used is the sample standard deviation, which is defined by Standardized testing is: in theory: a tool to ensure that student knowledge and aptitude in a given subject are examined with the same criteria across different schools. ...

 s = sqrt{frac{1}{N-1} sum_{i=1}^N (x_i - overline{x})^2},,

where scriptstyle{x_1,,x_2,,ldots,,x_N} is the sample and scriptstyleoverline{x} is the mean of the sample. The denominator N − 1 is the number of degrees of freedom in the vector scriptstyle(x_1-overline{x},,dots,,x_N-overline{x}). This article or section is in need of attention from an expert on the subject. ...


The reason for this definition is that s2 is an unbiased estimator for the variance σ2 of the underlying population, if that variance exists and the sample values are drawn independently with replacement. However, s is not an unbiased estimator for the standard deviation σ; it tends to underestimate the population standard deviation. Although an unbiased estimator for σ is known when the random variable is normally distributed, the formula is complicated and amounts to a minor correction. Moreover, unbiasedness, in this sense of the word, is not always desirable; see bias of an estimator. In statistics, a biased estimator is one that for some reason on average over- or underestimates what is being estimated. ... This article is about mathematics. ... The standard deviation is often estimated from a random sample drawn from the population. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... This article is about bias of statistical estimators. ...


Another estimator sometimes used is the similar expression

 sqrt{frac{1}{N} sum_{i=1}^N (x_i - overline{x})^2},,.

This form has a uniformly smaller mean squared error than does the unbiased estimator, and is the maximum-likelihood estimate when the population is normally distributed. In statistics the mean squared error of an estimator T of an unobservable parameter θ is i. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...


Standard deviation of a continuous random variable

Continuous distributions usually give a formula for calculating the standard deviation as a function of the parameters of the distribution. In general, the standard deviation of a continuous random variable X with probability density function p(x) is In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

sigma = sqrt{int (x-mu)^2 , p(x) , dx}

Where

mu = int x , p(x) , dx

Properties of Standard deviation

stdev(X + c) = Stdev(X) with c constant


stdev(cX)= c stdev(X)


stdev(X + Y) = SQRT(var(X)+var(Y)+2 covar(X,Y))


Example

We will show how to calculate the standard deviation of a population. Our example will use the ages of four young children: { 5, 6, 8, 9 }.


Step 1. Calculate the arithmetic mean overline{x}: In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

overline{x}=frac{1}{N}sum_{i=1}^N x_i

We have N = 4 because there are four data points:

x_1 = 5 quad x_2 = 6 quad x_3 = 8 quad x_4 = 9
overline{x}=frac{1}{4}sum_{i=1}^4 x_i       Substitute N=4
overline{x}=frac{1}{4} left ( x_1 + x_2 + x_3 +x_4 right )
overline{x}=frac{1}{4} left ( 5 + 6 + 8 + 9 right )
overline{x}= 7

Step 2. Calculate the standard deviation, sigma,!. (Since the four values represent the entire population, we do not use the formula for estimated standard deviation in this case):

sigma = sqrt{frac{1}{N} sum_{i=1}^N (x_i - overline{x})^2}
sigma = sqrt{frac{1}{4} sum_{i=1}^4 (x_i - 7)^2}       Substitute overline{x}=7 and N=4
sigma = sqrt{frac{1}{4} left [ (x_1 - 7)^2 + (x_2 - 7)^2 + (x_3 - 7)^2 + (x_4 - 7)^2 right ] }
sigma = sqrt{frac{1}{4} left [ (5 - 7)^2 + (6 - 7)^2 + (8 - 7)^2 + (9 - 7)^2 right ] }
sigma = sqrt{frac{1}{4} left ( (-2)^2 + (-1)^2 + 1^2 + 2^2 right ) }
sigma = sqrt{frac{1}{4} left ( 4 + 1 + 1 + 4 right ) }
sigma = sqrt{frac{10}{4}}
sigma = sqrt{2.5} approx 1.58

So the standard deviation of the ages of the four children is the square root of 2.5, or approximately 1.58.


Were this set a sample drawn from a larger population of children, and the question at hand was an estimate of the standard deviation of the population, convention would replace the denominator N (or 4) in step 2 here with N−1 (or 3).


Interpretation and application

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.


For example, each of the three data sets {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 set has a much smaller standard deviation than the other two because its values are all close to 7. In a loose sense, the standard deviation tells us how far from the mean the data points tend to be. It will have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of four siblings in years, the standard deviation is 5 years.


As another example, the data set {1000, 1006, 1008, 1014} 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 measurements should give 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. Measurement is the estimation of the magnitude of some attribute of an object, such as its length or weight, relative to a unit of measurement. ... “Accuracy” redirects here. ... In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. ...


Real-life 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).


Weather

As a simple example, consider average temperatures for cities. While two cities may each have an average temperature of 60 °F, it's helpful to understand that the range for cities near the coast is smaller than for cities inland, which clarifies that, while the average is similar, the chance for variation is greater inland than near the coast. For other uses, see Fahrenheit (disambiguation). ...


So, an average of 60 occurs for one city with highs of 80 °F and lows of 40 °F, and also occurs for another city with highs of 65 and lows of 55. The standard deviation allows us to recognize that the average for the city with the wider variation, and thus a higher standard deviation, will not offer as reliable a prediction of temperature as the city with the smaller variation and lower standard deviation.


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 be pretty good in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent they might be. So, a team that is consistently bad in most categories will have a low standard deviation. A team that is consistently good in most categories will also have a low standard deviation. A team with a high standard deviation might be the type of team that scores a lot (strong offense) but also concedes a lot (weak defense), or, vice versa, that might have a poor offense but compensates by being difficult to score on—teams with a higher standard deviation will be more unpredictable.


Trying to predict which teams, on any given day, will win, may include looking at the standard deviations of the various team "stats" ratings, in which anomalies can match strengths vs. weaknesses to attempt to understand what factors may prevail as stronger indicators of eventual scoring outcomes.


In racing, a driver is timed on successive laps. A driver with a low standard deviation of lap times is more consistent than a driver with a higher standard deviation. This information can be used to help understand where opportunities might be found to reduce lap times.


Finance

In finance, standard deviation is a representation of the risk associated with a given security (stocks, bonds, property, etc.), or the risk of a portfolio of securities. 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. The overall concept of risk is that as it increases, the expected return on the asset will increase as a result of the risk premium earned – in other words, investors should expect a higher return on an investment when said investment carries a higher level of risk.


For example, you have a choice between two stocks: Stock A historically returns 5% with a standard deviation of 10%, while Stock B returns 6% and carries a standard deviation of 20%. On the basis of risk and return, an investor may decide that Stock A is the better choice, because Stock B's additional percentage point of return generated (an additional 20% in dollar terms) is not worth double the degree of risk associated with Stock A. Stock B is likely to fall short of the initial investment more often than Stock A under the same circumstances, and will return only one percentage point more on average. In this example, Stock A has the potential to earn 10% more than the expected return, but is equally likely to earn 10% less than the expected return.


Calculating the average return (or arithmetic mean) of a security over a given number of periods will generate an expected return on the asset. For each period, subtracting the expected return from the actual return results in the variance. Square the variance in each period to find the effect of the result on the overall risk of the asset. The larger the variance in a period, the greater risk the security carries. Taking the average of the squared variances results in the measurement of overall units of risk associated with the asset. Finding the square root of this variance will result in the standard deviation of the investment tool in question.


Geometric interpretation

To gain some geometric insights, we will start with a population of three values, x1, x2, x3. This defines a point P = (x1, x2, x3) in R3. Consider the line L = {(r, r, r) : r in R}. 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. And that is indeed the case. Moving orthogonally from P to the line L, one hits the point:

R = (overline{x},overline{x},overline{x})

whose coordinates are the mean of the values we started out with. A little algebra shows that the distance between P and R (which is the same as the distance between P and the line L) is given by σ√3. An analogous formula (with 3 replaced by N) is also valid for a population of N values; we then have to work in RN.


Rules for normally distributed data

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for 68.27 % of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 %; three standard deviations (light, medium, and dark blue) account for 99.73 %; and four standard deviations account for 99.994 %.
Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for 68.27 % of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 %; three standard deviations (light, medium, and dark blue) account for 99.73 %; and four standard deviations account for 99.994 %.

In practice, one often assumes that the data are from an approximately normally distributed population. This is ideally justified by the classical central limit theorem, which says that sums of many independent, identically distributed random variables tend towards the normal distribution as a limit. If that assumption is justified, then about 68 % of the values are within 1 standard deviation of the mean, about 95 % of the values are within two standard deviations and about 99.7 % lie within 3 standard deviations. This is known as the 68-95-99.7 rule, or the empirical rule. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... ...


The confidence intervals are as follows: In statistics, a confidence interval (CI) is an interval estimate of a population parameter. ...

σ 68.26894921371%
95.44997361036%
99.73002039367%
99.99366575163%
99.99994266969%
99.99999980268%
99.99999999974%

For normal distributions, the two points of the curve which are one standard deviation from the mean are also the inflection points. Plot of y = x3 with inflection point of (0,0). ...


Chebyshev's inequality

Chebyshev's inequality proves that in any data set, nearly all of the values will be close to the mean value, where the meaning of "close to" is specified by the standard deviation. Chebyshev's inequality entails that for (nearly) all random distributions, not just normal ones, we have the following weaker bounds: In probability theory, Chebyshevs inequality (also known as Tchebysheffs inequality, Chebyshevs theorem, or the Bienaymé-Chebyshev inequality), named after Pafnuty Chebyshev, who first proved it, states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a...

At least 50% of the values are within √2 standard deviations from the mean.
At least 75% of the values are within 2 standard deviations from the mean.
At least 89% of the values are within 3 standard deviations from the mean.
At least 94% of the values are within 4 standard deviations from the mean.
At least 96% of the values are within 5 standard deviations from the mean.
At least 97% of the values are within 6 standard deviations from the mean.
At least 98% of the values are within 7 standard deviations from the mean.

And in general:

At least (1 − 1/k2) × 100% of the values are within k standard deviations from the mean.

Relationship between standard deviation and mean

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion 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 x1, ..., xn are real numbers and define the function: In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ...

sigma(r) = sqrt{frac{1}{N-1} sum_{i=1}^N (x_i - r)^2}

Using calculus, or simply by completing the square, it is possible to show that σ(r) has a unique minimum at the mean: For other uses, see Calculus (disambiguation). ... Completing the square is an algebra technique, also used in many types of calculus. ...

r = overline{x}.,

(This can also be done with fairly simple algebra alone, since σ2(r) is equated to a quadratic polynomial).


The coefficient of variation of a sample is the ratio of the standard deviation to the mean. It is a dimensionless number that can be used to compare the amount of variance between populations with different means. In probability theory and statistics, the coefficient of variation (CV) is a measure of dispersion of a probability distribution. ... In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ...


Rapid calculation methods

A slightly faster (significantly for running standard deviation) way to compute the population standard deviation is given by the following formula (though considerations must be made for round-off error, arithmetic overflow, and arithmetic underflow conditions): A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. ... The term arithmetic overflow or simply overflow has the following meanings. ... The term arithmetic underflow or simply underflow has the following meanings. ...

 sigma = sqrt{left(frac{1}{N}sum_{i=1}^N{{x_i}^2}right) - overline{x}^2} = sqrt{left(frac{1}{N}sum_{i=1}^N{{x_i}^2}right) - left(frac{1}{N}sum_{i=1}^N{{x_i}}right)^2} = frac{1}{N}sqrt{Nleft(sum_{i=1}^N{{x_i}^2}right) - left(sum_{i=1}^N{{x_i}}right)^2}

or

sigma= frac{1}{s_0}sqrt{s_0s_2-s_1^2}

where the power sums s0, s1, s2 are defined by

 s_j=sum_{k=1}^N{x_k^j}.

Similarly for sample standard deviation:

 s = sqrt{frac{sum_{i=1}^N{{x_i}^2} - Noverline{x}^2}{N-1} }.

Or from running sums:

 s = sqrt{frac{Nsum_{i=1}^N{{x_i}^2} - left(sum_{i=1}^N{x_i}right)^2}{N(N-1)}}.

See also algorithms for calculating variance. Algorithms for calculating variance play a minor role in statistical computing. ...


References

  1. ^ Sir Francis Galton discovered the standard deviation

See also

“Accuracy” redirects here. ... Algorithms for calculating variance play a minor role in statistical computing. ... For probability distributions having an expected value and a median, the mean (i. ... In probability theory, Chebyshevs inequality (also known as Tchebysheffs inequality, Chebyshevs theorem, or the Bienaymé-Chebyshev inequality), named after Pafnuty Chebyshev, who first proved it, states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a... In statistics, a confidence interval (CI) is an interval estimate of a population parameter. ... // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... In mathematics and statistics, deviation is a measure of difference for interval and ratio variables between the observed value and the mean. ... The geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... The mean absolute error is a term used in mathematics and error analysis similar to variance. ... This article is about mathematical mean. ... Pooled standard deviation is a way to find a better estimate of the true standard deviation given several different samples taken in different circumstances where the mean may vary between samples but the true standard deviation (precision) is assumed to remain the same. ... In statistics and data analysis, a raw score is an original datum that has not been transformed – for example, the original result obtained by a student on a test (i. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... The sample size of a statistical sample is the number of repeated measurements that constitute it. ... It has been suggested that this article or section be merged with Chromaticity. ... Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The standard error of a method of measurement or estimate is the estimated standard deviation of the error in that method. ... Compares the various grading methods in a normal distribution. ... The standard deviation is often estimated from a random sample drawn from the population. ... Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. ... The Yamartino method is an algorithm for calculating the standard deviation of wind direction () during a single pass through the incoming data. ...

External links

This article is about the field of statistics. ... Descriptive statistics are used to describe the basic features of the data in a study. ... This article is about mathematical mean. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... Look up range in Wiktionary, the free dictionary. ... This article is about mathematics. ... It has been suggested that this article or section be merged with inferential statistics. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. ... The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... In statistics, the Alternative Hypothesis is the hypothesis proposed to explain a statistically significant difference between results, that is if the Null Hypothesis has been rejected. ... Type I errors (or α error, or false positive) and type II errors (β error, or a false negative) are two terms used to describe statistical errors. ... The Z-test is a statistical test used in inference. ... A t-test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... Compares the various grading methods in a normal distribution. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ... In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ... A meta-analysis is a statistical practice of combining the results of a number of studies. ... Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ... The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. ... The Kaplan-Meier estimator (also known as the Product Limit Estimator) estimates the survival function from life-time data. ... The logrank test (sometimes called the Mantel-Haenszel test or the Mantel-Cox test) [1] is a hypothesis test to compare the survival distributions of two samples. ... Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. ... // Proportional hazards models are a sub-class of survival models in statistics. ... Several sets of (x, y) points, with the correlation coefficient of x and y for each set. ... In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no logical connection, yet it may be implied that they do, due to a certain third, unseen factor (referred to as a confounding factor or lurking variable). The spurious relationship gives an... In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ... In statistics, rank correlation is the study of relationships between different rankings on the same set of items. ... In statistics, Spearmans rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about... The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendalls Ï„ or Tau test(s)) is used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. ... In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ... In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term ε. The model can be written as Example of linear regression with one dependent and one independent variable. ... dataset with approximating polynomials Nonlinear regression in statistics is the problem of fitting a model to multidimensional x,y data, where f is a nonlinear function of x with parameters θ. In general, there is no algebraic expression for the best-fitting parameters, as there is in linear regression. ... Logistic regression is a statistical regression model for Bernoulli-distributed dependent variables. ...

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Standard Deviation and Variance: Common Measures of Variability (305 words)
The standard deviation formula is very simple: it is the square root of the variance.
An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score.
The standard deviation has proven to be an extremely useful measure of spread in part because it is mathematically tractable.
Standard deviation - Wikipedia, the free encyclopedia (1534 words)
Standard deviation is the most common measure of statistical dispersion, measuring how spread out the values in a data set are.
The standard deviation is defined as the square root of the variance.
The term standard deviation was introduced to statistics by Karl Pearson (On the dissection of asymmetrical frequency curves, 1894).
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