# Encyclopedia

.
A plot of a normal distribution (or bell curve).
^ When graphed, normally distributed data form the classic bell curve.
• Online Focus Group, Market Research » Blog Archive » Standard Deviation 28 January 2010 0:54 UTC iresearch.com [Source type: FILTERED WITH BAYES]

^ Randomly dropping balls create a normal distribution curve (bell curve) .

^ Normal distribution curve (bell curve) .

Each colored band has a width of one standard deviation.
A data set with a mean of 50 (shown in blue) and a standard deviation (σ) of 20.
.In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance.^ Then the average of the squared deviations is calculated, to give a result caused the variance and standard deviation is the square root of this variance.
• UltimaCalc, Doing Standard Deviation with UltimaCalc 28 January 2010 0:54 UTC www.iconico.com [Source type: FILTERED WITH BAYES]

^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

.Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation.^ Why the standard deviation is a good measure of the average distance to the mean.
• UsableStats: Statistics Tutorials 28 January 2010 0:54 UTC www.usablestats.com [Source type: FILTERED WITH BAYES]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

.It shows how much variation there is from the "average" (mean).^ Basically it's a measure of variability, or how much the price varies from its moving average.
• Standard Deviation 28 January 2010 0:54 UTC chartfilter.com [Source type: FILTERED WITH BAYES]

^ How much variation is there in the estimates of "typical" described above?

^ In simple terms, it shows how much variation there is from the "average" (mean).
• Standard Deviation: GMAT Math - Business Exchange 28 January 2010 0:54 UTC bx.businessweek.com [Source type: FILTERED WITH BAYES]

.A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.^ The standard deviation measures the spread of the data about the mean value.
• Standard Deviation 28 January 2010 0:54 UTC www.gcseguide.co.uk [Source type: FILTERED WITH BAYES]
• www.Garmento.org - Standard Deviation 28 January 2010 0:54 UTC www.garmento.org [Source type: FILTERED WITH BAYES]

^ The size of the mean value of the data set depends on the size of the standard deviation.
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Add a point between mean and mean + standard deviation.
• Mean, Variance, and Standard Deviation 28 January 2010 0:54 UTC www.fmi.uni-sofia.bg [Source type: Reference]

.For example, the average height for adult men in the United States is about 70 inches (178 cm), with a standard deviation of around 3 in (8 cm).^ Forget about the pooled standard deviation!
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ Example standard-deviation function .

^ This question about " Standard Deviation?
• Standard Deviation? - Yahoo! UK & Ireland Answers 28 January 2010 0:54 UTC uk.answers.yahoo.com [Source type: General]
• Standard Deviation? - Yahoo! Answers 28 January 2010 0:54 UTC answers.yahoo.com [Source type: General]

.This means that most men (about 68 percent, assuming a normal distribution) have a height within 3 in (8 cm) of the mean (67–73 in (170–185 cm)) – one standard deviation, whereas almost all men (about 95%) have a height within 6 in (15 cm) of the mean (64–76 in (163–193 cm)) – 2 standard deviations.^ For data that have a normal distribution, about 68 per cent of the data points fall within (plus or minus) one standard deviation from the mean and about 95 per cent fall within (plus or minus) two standard deviations.
• Dorlands Medical Dictionary:standard deviation 28 January 2010 0:54 UTC www.mercksource.com [Source type: Academic]

^ A normal distribution has a bell-shaped density curve described by its mean and standard deviation .
• The Normal Distribution 28 January 2010 0:54 UTC www.stat.yale.edu [Source type: Academic]

^ Is it possible for both the minimum and maximum values to be within one standard deviation of the mean?
• Mean, Variance, and Standard Deviation 28 January 2010 0:54 UTC www.fmi.uni-sofia.bg [Source type: Reference]

.If the standard deviation were zero, then all men would be exactly 70 in (178 cm) high.^ Which standard deviation formula would I use and where?
• Which standard deviation formula would I use and where? - Yahoo! UK & Ireland Answers 28 January 2010 0:54 UTC uk.answers.yahoo.com [Source type: General]

^ Investopedia Commentary A volatile stock would have a high standard deviation.
• Standard deviation Definition | Definition of Standard deviation at Dictionary.com 28 January 2010 0:54 UTC dictionary.reference.com [Source type: Reference]

^ If everyone were exactly the same, the standard deviation would be zero.
• Article: Mutual fund risk: standard deviation is not the answer: don't use standard deviation by itself as a measure of risk. It is misleading and wrong. - Money Digest | HighBeam Research - FREE trial 28 January 2010 0:54 UTC www.highbeam.com [Source type: Academic]
• Article: Standard deviation: a risky measurement tool: standard deviation measures the volatility of a mutual fund, but is imperfect as a risk measurement tool. - Money Digest | HighBeam Research - FREE trial 28 January 2010 0:54 UTC www.highbeam.com [Source type: Academic]

.If the standard deviation were 20 in (51 cm), then men would have much more variable heights, with a typical range of about 50 to 90 in (127 to 229 cm).^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

.Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal (bell-shaped).^ Find the sample variance and standard deviation.
• Sample Variance and Sample Standard Deviation 28 January 2010 0:54 UTC www.castleton.edu [Source type: Academic]

^ A normal distribution has a bell-shaped density curve described by its mean and standard deviation .
• The Normal Distribution 28 January 2010 0:54 UTC www.stat.yale.edu [Source type: Academic]

^ Also, how do u know when to find the population standard deviation or the sample standard deviation?
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

.In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ The standard deviation is the square root of the variance.  .
• Statistics in C# : variance, standard deviation, covariance & pearson 28 January 2010 0:54 UTC www.noviway.com [Source type: Academic]

.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.^ It will also calculate the standard errors of the mean, median, standard deviation, variance, an of the coefficient of variation.
• UltimaCalc, Doing Standard Deviation with UltimaCalc 28 January 2010 0:54 UTC www.iconico.com [Source type: FILTERED WITH BAYES]

^ The student will calculate the standard deviation and percentage error of the gathered temperatures.
• Using Standard Deviation, part 2 28 January 2010 0:54 UTC teachertech.rice.edu [Source type: FILTERED WITH BAYES]

^ To see an example of how this parameter is calculated go to Standard Deviation Calculation .

.The reported margin of error is typically about twice the standard deviation – the radius of a 95% confidence interval.^ Forget about the pooled standard deviation!
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ We typically calculate the confidence interval for the mean but it can be done for the estimate of the standard deviation as well.
• Confidence and Deviation : Eoin Lawless 28 January 2010 0:54 UTC blogs.sun.com [Source type: FILTERED WITH BAYES]

^ This note explains the formulae for the standard error and 95% confidence interval for the estimated within subject standard deviation, s w .
• What is the standard error of the within-subject standard deviation? 28 January 2010 0:54 UTC www-users.york.ac.uk [Source type: Academic]

.In science, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from causal variation.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ The standard deviation is the square root of the variance.  .
• Statistics in C# : variance, standard deviation, covariance & pearson 28 January 2010 0:54 UTC www.noviway.com [Source type: Academic]

.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.^ Standard deviation is used to measure the volatility of a mutual fund.
• Article: Standard deviation: a risky measurement tool: standard deviation measures the volatility of a mutual fund, but is imperfect as a risk measurement tool. - Money Digest | HighBeam Research - FREE trial 28 January 2010 0:54 UTC www.highbeam.com [Source type: Academic]
• Standard deviation: a risky measurement tool: standard deviation measures the volatility of a mutual fund, but is imperfect as a risk measurement tool. - Money Digest | Encyclopedia.com 28 January 2010 0:54 UTC www.encyclopedia.com [Source type: Academic]

^ So, investment gurus DEFINE volatility as the Standard Deviation.
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

^ Additionally, it will now return the standard deviation.
• Adding Standard Deviation to LINQ — Developer.com 28 January 2010 0:54 UTC www.developer.com [Source type: FILTERED WITH BAYES]
• Adding Standard Deviation to LINQ — Developer.com 28 January 2010 0:54 UTC www.developer.com [Source type: FILTERED WITH BAYES]

.The term standard deviation was first used[1] in writing by Karl Pearson[2] in 1894, following his use of it in lectures.^ Which standard deviation formula would I use and where?
• Which standard deviation formula would I use and where? - Yahoo! UK & Ireland Answers 28 January 2010 0:54 UTC uk.answers.yahoo.com [Source type: General]

^ Why use n-1 when calculating a standard deviation?
•  Why use n-1 when calculating a standard deviation? 28 January 2010 0:54 UTC www.graphpad.com [Source type: FILTERED WITH BAYES]

^ The formula for Standard Deviation is as follows: .
• Chapter 9 Standard Deviation 28 January 2010 0:54 UTC www.800score.com [Source type: FILTERED WITH BAYES]
• Chapter 9 Standard Deviation 28 January 2010 0:54 UTC www.800score.com [Source type: FILTERED WITH BAYES]

.This was as a replacement for earlier alternative names for the same idea: for example Gauss used "mean error".[3] A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ It will also calculate the standard errors of the mean, median, standard deviation, variance, an of the coefficient of variation.
• UltimaCalc, Doing Standard Deviation with UltimaCalc 28 January 2010 0:54 UTC www.iconico.com [Source type: FILTERED WITH BAYES]

.Note, however, that for measurements with percentage as unit, the standard deviation will have percentage points as unit.^ Standard deviation measures how widely spread data points are.
• Definition of standard_deviation - Chemistry Dictionary 28 January 2010 0:54 UTC www.chemicool.com [Source type: Academic]

^ The key point is that the standard deviation is an objective measure of variation.
• What is a standard deviation and how do I compute it? 28 January 2010 0:54 UTC www.winspc.com [Source type: FILTERED WITH BAYES]

^ However, the standard deviation will stay the same!
• Further Aspects of Standard Deviation 28 January 2010 0:54 UTC richardbowles.tripod.com [Source type: FILTERED WITH BAYES]

.When only a sample of data from a population is available, the population standard deviation can be estimated by a modified quantity called the sample standard deviation, explained below.^ Solving for sample standard deviation.
• Statistics Standard Deviation Equations Formulas Calculator - Sample 28 January 2010 0:54 UTC www.ajdesigner.com [Source type: Academic]

^ Statistically speaking, the standard deviation depends on whether you have sample data or population data.
• Java - Computing Standard Deviation | DreamInCode.net 28 January 2010 0:54 UTC www.dreamincode.net [Source type: General]

^ The sample standard deviation (usually represented by S) measures the variability of data in a sample.
• Standard Deviation Formula? - Yahoo! Answers 28 January 2010 0:54 UTC answers.yahoo.com [Source type: General]

## Basic example

Consider a population consisting of the following values:
$2,\;4,\;4,\;4,\;5,\;5,\;7,\;9.$
There are eight data points in total, with a mean (or average) value of 5:
$\frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5.$
To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result:
$\begin{array}{ll} (2-5)^2 = (-3)^2 = 9 & (5-5)^2 = 0^2 = 0 \ (4-5)^2 = (-1)^2 = 1 & (5-5)^2 = 0^2 = 0 \ (4-5)^2 = (-1)^2 = 1 & (7-5)^2 = 2^2 = 4 \ (4-5)^2 = (-1)^2 = 1 & (9-5)^2 = 4^2 = 16 \end{array}$
Next divide the sum of these values by the number of values and take the square root to give the standard deviation:
$\sqrt{\frac{9+1+1+1+0+0+4+16}{8}} = 2.$
Therefore, the above has a population standard deviation of 2.
The above assumes a complete population. .If the 8 values are obtained by random sampling from some parent population, then computing the sample standard deviation would use a denominator of 7 instead of 8. See the section Estimation below for an explanation.^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Part of the assignment is to compute the standard deviation for each sample that is generated.
• Re: standard deviation / limit problem 28 January 2010 0:54 UTC library.thinkquest.org [Source type: Original source]

^ Compute the sample standard deviation of the adjusted scores.
• How to Compute the Population Variance and Standard Deviation, find Simple Random Samples and Compare the Standard Deviations (Section 3.2 topic).doc 28 January 2010 0:54 UTC co-5.college-online.com [Source type: Academic]

## Definition

### Probability distribution or random variable

Let X be a random variable with mean value μ:
$\operatorname{E}[X] = \mu.\,\!$
.Here the operator E denotes the average or expected value of X.^ Answer #3: Standard deviation is a value that tells us how far above or below his average you can expect a player to be around 68% of the time.

^ The mean of a probability distribution is its average or expected value.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ The standard deviation is analyzed from two angles: 1) a probability parameter of binomial events (compared to the expected value); 2) a statistical parameter of a numerical series (compared to the mean average).
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

Then the standard deviation of X is the quantity
$\sigma = \sqrt{\operatorname{E}\left[(X - \mu)^2\right]}.$
.That is, the standard deviation σ (sigma) is the square root of the average value of (X − μ)2.^ The National Institute of Standards and Technology’s statistics Web site offers Critical Values of the Chi-Square Distribution for 1 through 100 degrees of freedom.
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ The positive square root of the variance, called the standard deviation, has the same units as the original variable and can be easier to interpret for this reason.
• RF Cafe - Variance and Standard & Mean Deviation 28 January 2010 0:54 UTC www.rfcafe.com [Source type: Academic]

^ Then the average of the squared deviations is calculated, to give a result caused the variance and standard deviation is the square root of this variance.
• UltimaCalc, Doing Standard Deviation with UltimaCalc 28 January 2010 0:54 UTC www.iconico.com [Source type: FILTERED WITH BAYES]

In the case where X takes random values from a finite data set $x_1, x_2, \ldots, x_N$, with each value having the same probability, the standard deviation is
$\sigma = \sqrt{\frac{(x_1-\mu)^2 + (x_2-\mu)^2 + \cdots + (x_N - \mu)^2}{N}},$
or, using summation notation,
$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}.$
.The standard deviation of a (univariate) probability distribution is the same as that of a random variable having that distribution.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ I think the same can be done for standard deviation as well.

^ The standard deviation for a discrete variable is defined as .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

.Not all random variables have a standard deviation, since these expected values need not exist.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ The standard deviation for a discrete variable is defined as .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

.For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined because its expected value is undefined.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

### Continuous random variable

The standard deviation of a continuous real-valued random variable X with probability density function p(x) is
$\sigma = \sqrt{\int (x-\mu)^2 \, p(x) \, dx}\,,$
where
$\mu = \int x \, p(x) \, dx\,,$
and where the integrals are definite integrals taken for x ranging over the sample space of X.
.In the case of a parametric family of distributions, the standard deviation can be expressed in terms of the parameters.^ Other distributions have standard deviations, too.
• Alternatives to Standard Deviation as Risk Measure | Bionic Turtle 28 January 2010 0:54 UTC www.bionicturtle.com [Source type: FILTERED WITH BAYES]

^ Normal distribution is knoown as two parameter distribution - standard deviation is an independant parameter.
• Re: standard deviation / limit problem 28 January 2010 0:54 UTC library.thinkquest.org [Source type: Original source]

^ Z is expressed in terms of the number of standard deviations from the mean value.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

.For example, in the case of the log-normal distribution with parameters μ and σ2, the standard deviation is [(exp(σ2)-1)exp(2μ+σ2)]1/2.^ Cautions: The tests on standard deviation or variance of a population require that the underlying population must be normal .
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ The normal distribution has the following characteristics: 68.3% of the area lies between x̄ - 1s and x̄ + 1s 95.5% of the area lies between x̄ - 2s and x̄ + 2s 99.7% of the area lies between x̄ - 3s and x̄ + 3s For example, if a certain control gives a mean test result of 5.6 with standard deviation .8, then 95.5% of future tests on that control will be in the range 4.0 - 7.2, within 2 standard deviations of the mean.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ In this case the population standard deviation gives the absurd result σ x =0, whereas the sample standard deviation is undefined (0/0); a proper representation of the ignorance of the standard deviation after just one measurement.
• standard deviation@Everything2.com 28 January 2010 0:54 UTC everything2.com [Source type: FILTERED WITH BAYES]
• standard deviation@Everything2.com 28 January 2010 0:54 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

## 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.^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Solution : First, find the sample standard deviation, which is 6.42 g.
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ Finding the standard deviation of a merged group .
• Further Aspects of Standard Deviation 28 January 2010 0:54 UTC richardbowles.tripod.com [Source type: FILTERED WITH BAYES]

.In cases where that cannot be done, the standard deviation σ is estimated by examining a random sample taken from the population.^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ To estimate the standard deviation σ of a population at confidence level 1−α, the bounds are .
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ Sample standard deviation.
• Standard Deviation Formula? - Yahoo! Answers 28 January 2010 0:54 UTC answers.yahoo.com [Source type: General]

Some estimators are given below:

### With standard deviation of the sample

An estimator for σ sometimes used is the standard deviation of the sample, denoted by sn and defined as follows:
$s_n = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}.$
.This estimator has a uniformly smaller mean squared error than the "sample standard deviation" (see below), and is the maximum-likelihood estimate when the population is normally distributed.^ From Karen: How do you compare to see if a sample standard deviation is different than the population standard deviation?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]

^ Standard deviation is the square root of the average squared deviation from the mean.

^ The standard deviation is the square root of that.
• Standard deviation of rolling dice - GameDev.Net Discussion Forums 28 January 2010 0:54 UTC www.gamedev.net [Source type: General]

.But this estimator, when applied to a small or moderately-sized sample, tends to be too low: it is a biased estimator.^ Unfortunately, the PV-WAVE documentation does not say which distribution is used in its algorithm for small sample sizes, only that the confidence limits "assume normality".
• Visual Numerics Forums - confidence interval for standard deviation? 28 January 2010 0:54 UTC forums.vni.com [Source type: FILTERED WITH BAYES]
• Visual Numerics Forums - confidence interval for standard deviation? 28 January 2010 0:54 UTC forums.vni.com [Source type: FILTERED WITH BAYES]

^ As this situation very rarely arises, should be used and ignored, although even for moderate sample sizes the difference is going to be small.
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

^ For small sample sizes, the Chi Squared Test will not always produce an accurate probability.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

.The standard deviation of the sample is the same as the population standard deviation of a discrete random variable that can assume precisely the values from the data set, where the probability for each value is proportional to its multiplicity in the data set.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ Statistically speaking, the standard deviation depends on whether you have sample data or population data.
• Java - Computing Standard Deviation | DreamInCode.net 28 January 2010 0:54 UTC www.dreamincode.net [Source type: General]

### With sample standard deviation

The most common estimator for σ used is an adjusted version, the sample standard deviation, denoted by "s" and defined as follows:
$s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2},$
where $\scriptstyle\{x_1,\,x_2,\,\ldots,\,x_N\}$ is the sample and $\scriptstyle\overline{x}$ is the mean of the sample. This correction (the use of N − 1 instead of N) is known as Bessel's correction. .The reason for this correction 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.^ Sample estimate of population mean .
• AP* Statistics Tutorial: Standard Error 28 January 2010 0:54 UTC stattrek.com [Source type: Academic]

^ Equation (3) above is an unbias estimate of population variance.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Cautions: The tests on standard deviation or variance of a population require that the underlying population must be normal .
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

.However, s is not an unbiased estimator for the standard deviation σ; it tends to underestimate the population standard deviation.^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ To estimate the standard deviation σ of a population at confidence level 1−α, the bounds are .
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ Estimate the standard deviation of the population, with 95% confidence.
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

.Note that the term "standard deviation of the sample" is used for the uncorrected estimator (using N) whilst the term "sample standard deviation" is used for the corrected estimator (using N − 1).^ A standard error of a statistic (or estimator) is the (estimated) standard deviation of the statistic.
• Standard Deviation Scholarly Articles 28 January 2010 0:54 UTC www.hedgefund-index.com [Source type: FILTERED WITH BAYES]

^ Have a look here under # 1.2 Estimating standard deviation from a sample.
• Standard deviation of rolling dice - GameDev.Net Discussion Forums 28 January 2010 0:54 UTC www.gamedev.net [Source type: General]

^ Will correcting this error affect the standard deviation?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

.The denominator N − 1 is the number of degrees of freedom in the vector of residuals, $\scriptstyle(x_1-\overline{x},\,\dots,\,x_N-\overline{x})$.^ (The division by the number of observations minus one instead of the number of observations itself to obtain the mean square is because "degrees of freedom" must be used.
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

### With interquartile range

The statistic
$\frac ext{IQR}{1.349}$
.(1.349 is an approximation) where IQR is the interquartile range of the sample, is a consistent estimate of σ if the population is normally distributed.^ Most sample data are not normally distributed.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ The normal range for all tests is defined as plus or minus 2 SDs, which generally equates to 95% of the population.
• » Standard Deviation Buzz - DiabetesMine: the all things diabetes blog 28 January 2010 0:54 UTC www.diabetesmine.com [Source type: General]

^ From Fareeha: A population consists of six numbers 1,2,3,4,5,6 a) How many different samples of two numbers can be selected with replacement?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]

.The interquartile range IQR is the difference of the 3rd quartile of the data and the 1st quartile of the data.^ The range of a set of numerical data points is the difference between the largest value and the smallest value.
• Beginning Algebra Tutorial on Central Tendencies 28 January 2010 0:54 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Unlike range and quartiles, the variance combines all the values in a data set to produce a measure of spread.
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

.The asymptotic relative efficiency (ARE) of this estimator with respect to the one from sample standard deviation is 0.37. Hence, for normal data, it is better to use the one from sample standard deviation; when data is with thicker tails, this estimator can be more efficient.^ Answer #2: Standard Deviation is one number that we use to describe a bunch of numbers.

^ In a normal distribution, what percent of data would be more than +/- 3 standard deviations from the mean?
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ A standard error of a statistic (or estimator) is the (estimated) standard deviation of the statistic.
• Standard Deviation Scholarly Articles 28 January 2010 0:54 UTC www.hedgefund-index.com [Source type: FILTERED WITH BAYES]

[4]

### Other estimators

.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: see Unbiased estimation of standard deviation for more details.^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ Variance and Standard Deviation of a Beta Distribution If X has the beta distribution function f(x) = ( +1)( +2)x (1 x), then You can see the significance of the standard deviation quite clearly in the normal distribution.
• Calculus and Probability 3 28 January 2010 0:54 UTC www.hofstra.edu [Source type: Academic]

^ Upgrade to FORMULA.EXE: standard deviation, binomial distribution .
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]
• FORMULA software, standard deviation, politics 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

Moreover, unbiasedness, (in this sense of the word), is not always desirable: see bias of an estimator.

## Identities and mathematical properties

.The standard deviation is invariant to changes in location, and scales directly with the scale of the random variable.^ Variance and standard deviation Statistics Canada .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ Basic Rule of the Variance and the standard deviation .
• Duncan Williamson: Introduction to the Standard Devaition stdev.html 28 January 2010 0:54 UTC www.duncanwil.co.uk [Source type: FILTERED WITH BAYES]

^ The standard deviation for a discrete variable is defined as .
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

Thus, for a constant c and random variables X and Y:
$\operatorname{stdev}(X + c) = \operatorname{stdev}(X). \,$
$\operatorname{stdev}(cX) = |c|\,\operatorname{stdev}(X). \,$
The standard deviation of the sum of two random variables can be related to their individual standard deviations and the covariance between them:
$\operatorname{stdev}(X + Y) = \sqrt{\operatorname{var}(X) + \operatorname{var}(Y) + 2\operatorname{cov}(X,Y)}. \,$
where $\operatorname{var}$ and $\operatorname{cov}$ stand for variance and covariance, respectively.
.The calculation of the sum of squared deviations can be related to moments calculated directly from the data.^ Calculate statistics related to the standard deviation: Sums, mean average, and average deviation from the mean.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ It is calculated as the average squared deviation of each number from the mean of a data set.
• Learning Resources: Statistics: Power from Data! Variance and standard deviation 28 January 2010 0:54 UTC www.statcan.gc.ca [Source type: Reference]

^ The standard deviation (typically denoted with the Greek letter sigma, σ ) is the root of the average of the sum of the squares of the distances of each number from the average of all the numbers (yadda yadda yadda).
• Insyst - Filling In The Details 28 January 2010 0:54 UTC www.insystusa.com [Source type: Reference]

In general, we have
$\operatorname{stdev}(X) = \sqrt{E(X-EX)^2} = \sqrt{E(X^2) - (EX)^2}.$
For a finite population with equal probabilities on all points, we have
$\sqrt{\frac{1}{N}\sum_{i=1}^N(X_i-\overline{x})^2} = \sqrt{\frac{1}{N} \left(\sum_{i=1}^N x_i^2\right) - \overline{x}^2}.$
.Thus, the standard deviation is equal to the square root of (the average of the squares less the square of the average).^ The standard deviation is the square root of that.
• Standard deviation of rolling dice - GameDev.Net Discussion Forums 28 January 2010 0:54 UTC www.gamedev.net [Source type: General]

^ The positive square root of the variance is called the standard deviation of X and is denoted by .
• THE MEAN, VARIANCE, AND STANDARD DEVIATION 28 January 2010 0:54 UTC cnx.org [Source type: Academic]

^ Calculate the average of the squared variances and then calculate the square root of this average variance and you will get the standard deviation of the investment in question.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.See computational formula for the variance for a proof of this fact, and for an analogous result for the sample standard deviation.^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ It is a variant of a standard deviation mathematical model.
• Standard deviation downloads at VicMan 28 January 2010 0:54 UTC www.vicman.net [Source type: General]

^ To find the variance and standard deviation of X we first find .
• THE MEAN, VARIANCE, AND STANDARD DEVIATION 28 January 2010 0:54 UTC cnx.org [Source type: Academic]

## 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.^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ It means they score more points!

^ The mean is displayed by Shift and and the standard deviation by Shift and .
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

.For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively.^ One important consequence of these results is that if we have a collection of numbers, say {G k }, with Mean = 0 and SD = 1, then the collection {λ G k + C} will have Mean = C and Standard Deviation = λ.
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

^ Cautions: The tests on standard deviation or variance of a population require that the underlying population must be normal .
• Inferences about One Population Standard Deviation — MATH200 (TC3, Brown) 28 January 2010 0:54 UTC www.tc3.edu [Source type: FILTERED WITH BAYES]

^ Then the question asks you to calculate the mean and standard deviation of the students' ages on the 1st December 2000.
• Further Aspects of Standard Deviation 28 January 2010 0:54 UTC richardbowles.tripod.com [Source type: FILTERED WITH BAYES]

.The third population 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.^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

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 a population of four siblings in years, the standard deviation is 5 years.^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ However, the standard deviation involves every value of its data set.
• Beginning Algebra Tutorial on Central Tendencies 28 January 2010 0:54 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Standard deviation from ungrouped data .
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

.As another example, the population {1000, 1006, 1008, 1014} may represent the distances traveled by four athletes, measured in meters.^ As illustrated in the example above, most of the time it is infeasible to directly measure a population parameter.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ For example, an analyst may make four measurements upon a given production lot of material (population).
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.ivstandards.com [Source type: Academic]
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.inorganicventures.com [Source type: Academic]

.It has a mean of 1007 meters, and a standard deviation of 5 meters.^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ The mean is displayed by Shift and and the standard deviation by Shift and .
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

^ Comparing means and standard deviations .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

.Standard deviation may serve as a measure of uncertainty.^ If this analysis was repeated several times to produce several sample sets (four each) of data, it would be expected that each set of measurements would have a different mean and a different estimate of the standard deviation.
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.ivstandards.com [Source type: Academic]
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.inorganicventures.com [Source type: Academic]

^ The symbol for mean average is the Greek letter MU. The symbol of standard deviation is the Greek letter SIGMA. The variance (sigma square) is the measurement of the squared deviations.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ WM n ) 2 ( 4 ) ##EQU00003## [0017] (E) may include obtaining the variance data or the standard deviation data in accordance with the weighted averages by assuming that a horizontal axis indicates weighted averages and a vertical axis indicates variances or standard deviations.
• Method of obtaining variance data or standard deviation data for reducing noise, and digital photographing apparatus including recording medium storing variance data or standard deviation data for reducing noise - Patent application 28 January 2010 0:54 UTC www.faqs.org [Source type: Reference]

.In physical science, for example, the reported standard deviation of a group of repeated measurements should give the precision of those measurements.^ Data sets with a small standard deviation have tightly grouped, precise data.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ The variance gives rise to standard deviation.
• Z-5: Sum of Squares, Variance, and the Standard Error of the Mean - Westgard QC 28 January 2010 0:54 UTC www.westgard.com [Source type: Academic]

^ Standard deviation from grouped data .
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

.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.^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ The mean is displayed by Shift and and the standard deviation by Shift and .
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

^ Comparing means and standard deviations .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

.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.^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Susan pose la question : What is the range of values for standard deviation?

^ Range of values for standard deviation .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

See prediction interval.

### 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).^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Standard deviation is the square root of the average squared deviation from the mean.

^ Mean and the S.D. of a set values is 100 and 5.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

#### Weather

.As a simple example, consider average temperatures for cities.^ Average Calculation (within the annual returns table) The average displays a simple average of the displayed statistic; however partial years are considered within the calculation.
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

.While two cities may each have an average temperature of 15 °C, 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.^ The average score might be 100 and the two side bars represent + or -15%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ WM n ) 2 ( 4 ) ##EQU00003## [0017] (E) may include obtaining the variance data or the standard deviation data in accordance with the weighted averages by assuming that a horizontal axis indicates weighted averages and a vertical axis indicates variances or standard deviations.
• Method of obtaining variance data or standard deviation data for reducing noise, and digital photographing apparatus including recording medium storing variance data or standard deviation data for reducing noise - Patent application 28 January 2010 0:54 UTC www.faqs.org [Source type: Reference]

^ Thus, the noise may be partially reduced by using the graph of the averages in the horizontal axis versus the variances or the standard deviations in the vertical axis.
• Method of obtaining variance data or standard deviation data for reducing noise, and digital photographing apparatus including recording medium storing variance data or standard deviation data for reducing noise - Patent application 28 January 2010 0:54 UTC www.faqs.org [Source type: Reference]

.So, an average of 15 occurs for one city with highs of 25 °C and lows of 5 °C, and also occurs for another city with highs of 18 and lows of 12. 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.^ It is a variant of a standard deviation mathematical model.
• Standard deviation downloads at VicMan 28 January 2010 0:54 UTC www.vicman.net [Source type: General]

^ To find the variance and standard deviation of X we first find .
• THE MEAN, VARIANCE, AND STANDARD DEVIATION 28 January 2010 0:54 UTC cnx.org [Source type: Academic]

^ The Standard Deviation of these fifty-one annual returns is was 15.8%.
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

#### 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.^ If an explanation cannot be found for an outlier (other than it appears too high or low), there is a convenient test that can be used for the rejection of possible outliers from limited data sets.
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.ivstandards.com [Source type: Academic]
• Accuracy, Precision, Mean and Standard Deviation: ICP and ICP-MS Operations 28 January 2010 0:54 UTC www.inorganicventures.com [Source type: Academic]

^ Alternative Hypothesis: There is some other reason that they all received the same score.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ There is also plan to relieve teachers from some of the administrative tasks so that they can concentrate on the things they do best, teach.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

Chances are, the teams that lead in the standings will not show such disparity, but will perform well in most categories. .The lower the standard deviation of their ratings in each category, the more balanced and consistent they will tend to be.^ Stephanie Sollow (ProgressiveWorld.net) Rating: 5/5 "One of the highlights of 2002 is Standard Deviation, Electrum's second release.
• Net Dot Music, Inc. * Electrum - Standard Deviation * Reviews 28 January 2010 0:54 UTC www.netdotmusic.com [Source type: General]

^ It is true that the per year average rate of return has a smaller standard deviation for a longer time horizon.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ These numbers dont tell us who was better, they only indicate who was more CONSISTENT and who was more ERRATIC .

.Whereas, teams with a higher standard deviation will be more unpredictable.^ Focus on the standard deviation of the rate of return should NEVER obscure the more proper emphasis on the possible dollar values of a portfolio strategy".
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ So, if your mean blood sugar were 120 mg/dl, you would want your standard deviation to be no more than 40 mg/dl, or one-third of the mean.
• » Standard Deviation Buzz - DiabetesMine: the all things diabetes blog 28 January 2010 0:54 UTC www.diabetesmine.com [Source type: General]

^ A data series with a standard deviation less than the median is more desirable as it is more highly predictable in its future movements.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

.For example, a team that is consistently bad in most categories will have a low standard deviation.^ This example is actually population standard deviation.

^ Standard deviation is a most fundamental element of randomness.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ This procedure illustrates the structure of the standard deviation, in particular that the two extreme values 0.1 and 3.2 contribute most to the sum of the differences squared.
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

.A team that is consistently good in most categories will also have a low standard deviation.^ Standard deviation is a most fundamental element of randomness.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ This procedure illustrates the structure of the standard deviation, in particular that the two extreme values 0.1 and 3.2 contribute most to the sum of the differences squared.
• eBMJ -- Statistics at Square One: 2. Mean and standard deviation 28 January 2010 0:54 UTC www.bmj.com [Source type: FILTERED WITH BAYES]

^ Looking at a standard deviation will tell you if your average is made up of good numbers (great!
• Standard Deviation - Diabetes Daily 28 January 2010 0:54 UTC www.diabetesdaily.com [Source type: General]

.However, 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.^ Put into terms for securities, the average return might be 10% with a standard deviation (68% of the time) of 21%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ However, an analysis of extensive test data indicates that there is a negative relationship between attenuation performance and the standard deviation.
• Hearing protector performance and standard deviation Williams W, Dillon H - Noise Health 28 January 2010 0:54 UTC www.noiseandhealth.org [Source type: Academic]

^ Ideally, what testing should be done prior to starting a new lot of control material in order to establish its mean and standard deviation?
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

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.^ Put into terms for securities, the average return might be 10% with a standard deviation (68% of the time) of 21%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ It is true that the per year average rate of return has a smaller standard deviation for a longer time horizon.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Let's say we consider minus three standard deviations to be a big loss: the S&P 500 experienced a daily loss of minus three standard deviations about -3.4% of the time.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

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.^ DEFINE risk as the Standard Deviation ...
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

^ In the field of finance, standard deviation represents the risk associated with a security (stocks or bonds), or the risk of a portfolio of securities (including actively managed mutual funds, index mutual funds, or ETFs).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Put into terms for securities, the average return might be 10% with a standard deviation (68% of the time) of 21%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

), or the risk of a portfolio of securities .(actively managed mutual funds, index mutual funds, or ETFs).^ Actively managed funds sometimes outperform index funds.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Investing in any mutual fund, index or actively managed, does not guarantee that an investor will make money, avoid losing capital, or indicate that the investment is risk-free.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Individual stock prices, mutual fund manager performances, and short-term market performance tend not to be mean reverting.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.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).^ Risk is an important factor in determining how to efficiently manage investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (the basis for mean-variance optimization).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Where M RD = The mean return of the dependent variable .
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

^ Investing in any mutual fund, index or actively managed, does not guarantee that an investor will make money, avoid losing capital, or indicate that the investment is risk-free.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.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, or uncertainty of that return.^ The overall concept of risk is that as it increases, the expected return on the asset should increase as a result of the risk premium earned – in other words, investors should not expect a higher returns on an investment without that investment having a higher degree of risk, or uncertainty of those returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ When evaluating investments, investors should always estimate both the expected average return and the uncertainty of future returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns.^ When evaluating investments, investors should always estimate both the expected average return and the uncertainty of future returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ The overall concept of risk is that as it increases, the expected return on the asset should increase as a result of the risk premium earned – in other words, investors should not expect a higher returns on an investment without that investment having a higher degree of risk, or uncertainty of those returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.Standard deviation provides a quantified estimate of the uncertainty of future returns.^ Volatility is annualized standard deviation of returns.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Standard deviation provides a quantified estimate of the uncertainty of those future returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ This estimation of the standard deviation is calculated from the relationship: .
• 141 Standard Deviation Experiment 28 January 2010 0:54 UTC www.grossmont.edu [Source type: FILTERED WITH BAYES]

.For example, let's assume an investor had to choose between two stocks.^ For example, let's assume an investor had to choose between two stocks.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.Stock A over the last 20 years had an average return of 10%, with a standard deviation of 20 percentage points (pp) and Stock B, over the same period, had average returns of 12%, but a higher standard deviation of 30 pp.^ I think the same can be done for standard deviation as well.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Volatility is annualized standard deviation of returns.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ The average monthly return was 1.14% and the monthly standard deviation was 4.01%.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional 2% points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return).^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ Figure 8-1a Enlarge On the basis of risk and return, an investor may decide that The Walt Disney Company is the better choice, because Hewlett Packard's additional 2% points of return is not worth the additional 10% standard deviation (greater risk or uncertainty of the expected return).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Risk (denominator) is defined as the standard deviation of the investment returns.
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

.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 2% more on average.^ Hewlett Packard has a lower risk/reward ratio (0.333 for HP versus 0.385 for Disney) and is more likely to fall short of the initial investment under the same circumstances, and is estimated to return only 2% more on average.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ So with only 3 years of returns data on the S&P 500, the error in the average return is 2.6 times larger than having 20 years of data.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Annualized Tracking Error is measured by taking the square root of the average of the squared deviations between the investments returns and the benchmarks returns, then multiplying the result by the square root of 12.
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

.In this example, Stock A is expected to earn about 10%, plus or minus 20 pp (a range of 30% to -10%), about two-thirds of the future year returns.^ However, over 80 years, the rate of return of the S&P 500 has tended to average in the 10% range plus or minus 20% two-thirds of the time.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ In this example, The Walt Disney Company is expected to earn about 10%, plus or minus 26% (a range from 36% to -16%), about two-thirds of the future annual returns.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.When considering more extreme possible returns or outcomes in future, an investor should expect results of up to 10% plus or minus 60 pp, or a range from 70% to (−)50%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).^ Forget about the pooled standard deviation!
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Up 2 standard deviations for the day.
• Recent "Standard deviation breakout" Alerts 28 January 2010 0:54 UTC www.trade-ideas.com [Source type: General]

.Calculating the average return (or arithmetic mean) of a security over a given number of periods will generate an expected return on the asset.^ If you want to calculate the standard deviation of a certain stock or index, start by calculating the average return (or arithmetic mean) of the security over a given number of periods, like 20 years or more.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Put into terms for securities, the average return might be 10% with a standard deviation (68% of the time) of 21%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Mean reversion is a tendency for certain random variables to remain at or return over time to a long-run average level.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

.For each period, subtracting the expected return from the actual return results in the variance.^ For each year, subtract the average return from the actual return for that year, which will give you the "variances".
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ These results aren't from a historical example, but in terms of expectations, given a standard deviation of (variance is the square of standard deviation, ^2) and an expected average gain of , the expected annualized return is approximately - (^2 ÷ 2).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ It is the return generated with the pluses and minuses you might expect 68% of the time over a period of one year.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

.Square the variance in each period to find the effect of the result on the overall risk of the asset.^ Square the variance in each period.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Then, we'll calculate mean by using formula A. Next, we can find the difference from the mean for each result, and square that result for the two relevant columns.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ One way of quantifying this spread is with the standard deviation, which is denoted with s.To calculate the standard deviation, first find for each point, square the results, add them, divide by n-1, and finally take the square root.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

.The larger the variance in a period, the greater risk the security carries.^ The larger the variance in a period, the greater risk the security carries.
• Riskese - Index Funds Advisors, Inc. 28 January 2010 0:54 UTC www.ifa.com [Source type: FILTERED WITH BAYES]

^ Larger differences tend to arise at the lower values where the variance in attenuation values is greater (see below).
• Hearing protector performance and standard deviation Williams W, Dillon H - Noise Health 28 January 2010 0:54 UTC www.noiseandhealth.org [Source type: Academic]

.Taking the average of the squared variances results in the measurement of overall units of risk associated with the asset.^ The variance (sigma squared) is the measurement of the squared deviations.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ The third part examines the interpretation of the results generated by a risk analysis application including investment decision criteria and various measures of risk based on the expected value concept.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ The symbol for mean average is the Greek letter MU. The symbol of standard deviation is the Greek letter SIGMA. The variance (sigma square) is the measurement of the squared deviations.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

.Finding the square root of this variance will result in the standard deviation of the investment tool in question.^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ The variance (sigma squared) is the measurement of the squared deviations.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

.Population standard deviation is used to set the width of Bollinger bands, a widely adopted technical analysis tool.^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ From Shahzada: when and why we use standard deviation in Data analysis?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]

^ Solve for the standard deviation of the data set.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

For example, the upper Bollinger band is given as:
$\bar{x} + n \sigma_{x}. \,$

### 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.^ Absolute Value vs. Standard Deviation ?
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Range of values for standard deviation .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Expected value and standard deviation .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

.So it is not unreasonable to assume that the standard deviation is related to the distance of P to L.^ Calculate statistics related to the standard deviation: Sums, mean average, and average deviation from the mean.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ This author offers this type of evaluation for statistical data as related to the standard deviation.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

^ Here is a sample of calculating the standard deviation formula and related statistical parameters.
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

And that is indeed the case. To move orthogonally from L to the point P, one begins at the point:
$M = (\overline{x},\overline{x},\overline{x})$
whose coordinates are the mean of the values we started out with. .A little algebra shows that the distance between P and M (which is the same as the orthogonal distance between P and the line L) is equal to the standard deviation of the vector x1, x2, x3, divided by the square root of the number of dimensions of the vector.^ I think the same can be done for standard deviation as well.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Note that {Standard Deviation of the x's} 2 is the difference between the average of the squares and the square of the average .
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

^ To get the standard deviation, just take the square root of the resulting number .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

### Chebyshev's inequality

.An observation is rarely more than a few standard deviations away from the mean.^ Find the mean and the standard deviation.
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ Comparing means and standard deviations .
• Quandaries & Queries at Math Central 28 January 2010 0:54 UTC mathcentral.uregina.ca [Source type: FILTERED WITH BAYES]
• Dilemmes et doutes - Centrale des maths 28 January 2010 0:54 UTC centraledesmaths.uregina.ca [Source type: FILTERED WITH BAYES]

^ In a normal distribution, what percent of data would be more than +/- 3 standard deviations from the mean?
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

.Chebyshev's inequality entails the following bounds for all distributions for which the standard deviation is defined.^ DEFINE risk as the Standard Deviation ...
• Standard Deviation and Risk 28 January 2010 0:54 UTC www.gummy-stuff.org [Source type: FILTERED WITH BAYES]

^ Upgrade to FORMULA.EXE: standard deviation, binomial distribution .
• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

• Standard Deviation: Formula, Algorithm, Software 28 January 2010 0:54 UTC saliu.com [Source type: FILTERED WITH BAYES]

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.
And in general:
At least (1 − 1/k2) × .100% of the values are within k standard deviations from the mean.^ The values necessary for calculating the acceptable ranges are the mean and standard deviation.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ Z is expressed in terms of the number of standard deviations from the mean value.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Define a probability distribution by a sequence of control points, then watch how the mean and standard deviation of the distribution change as you move the points.
• Mean and Standard Deviation of a Distribution - Wolfram Demonstrations Project 28 January 2010 0:54 UTC demonstrations.wolfram.com [Source type: FILTERED WITH BAYES]

[5]

### Rules for normally distributed data

.
Dark blue is less than one standard deviation from the mean.
^ Stephanie Sollow (ProgressiveWorld.net) Rating: 5/5 "One of the highlights of 2002 is Standard Deviation, Electrum's second release.
• Net Dot Music, Inc. * Electrum - Standard Deviation * Reviews 28 January 2010 0:54 UTC www.netdotmusic.com [Source type: General]

^ Say we have a reactor with a mean pressure reading of 100 and standard deviation of 7 psig.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ In other words, you would prefer a standard deviation of 12% rather than a 10% return with a deviation of 20%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

.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 a normal distribution, what percent of data would be more than +/- 3 standard deviations from the mean?
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ Normal Distribution As stated before, the standard deviation is a measure of the spread of a distribution.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ That is the definition for two standard deviations.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

The two points of the curve which are one standard deviation from the mean are also the inflection points.
The central limit theorem says that the distribution of a sum of many independent, identically distributed random variables tends towards the famous "bell-shaped" normal distribution with a pdf of:
$\frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2} \right)$
where μ is the arithmetic mean of the sample. .The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though also appears in the normalizing constant to keep the distribution normalized for different widths.^ How to calculate standard deviation .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ How do i work out the standard deviation from this?
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ Normal Distribution As stated before, the standard deviation is a measure of the spread of a distribution.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

.If a data distribution is approximately normal then the proportion of data values within z standard deviations of the mean is defined by erf ( / √2).^ In a normal distribution, what percent of data would be more than +/- 3 standard deviations from the mean?
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ Table 7 displays the group means, standard deviations and the F-values.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

^ Normal Distribution As stated before, the standard deviation is a measure of the spread of a distribution.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

.The percentage of data values within z standard deviations of the mean is defined by erf( / √2) × 50% + 50%.^ Motives for teaching: percentage, means, standard deviation and rank order .
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

^ Table 7 displays the group means, standard deviations and the F-values.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

^ As defined by a Gaussian distribution curve, what percentage of values would be expected to fall within two standard deviations of the mean: .
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

.If a data distribution is approximately normal then about 68% of the data values are within 1 standard deviation of the mean (mathematically, μ ± σ, where μ is the arithmetic mean), about 95% are within two standard deviations (μ ± 2σ), and about 99.7% lie within 3 standard deviations (μ ± 3σ).^ At 2 sigma (two standard deviations) approximately 95.4% of your data will fall under the curve.
• Standard Deviation - Diabetes Daily 28 January 2010 0:54 UTC www.diabetesdaily.com [Source type: General]

^ Most sample data are not normally distributed.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ That is the definition for two standard deviations.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

.This is known as the 68-95-99.7 rule, or the empirical rule.^ Specifically: 68% of the area of the curve is within the range of μ ± 1σ 95% of the area of the curve is within the range of μ ± 2σ 99% of the area of the curve is within the range of μ ± 3σ Commit these numbers to memory: 68-95-99!
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ Then you can say that 68% of all data from the population will be within 1s of , 95% within 2s, and 99% within 3s.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

For various values of z, the percentage of values expected to lie in and outside the symmetric confidence interval CI = (−) are as follows:
zσ percentage within CI percentage outside CI ratio outside CI
68.2689492% 31.7310508% 1 / 3.1514871
1.645σ 90% 10% 1 / 10
1.960σ 95% 5% 1 / 20
95.4499736% 4.5500264% 1 / 21.977894
2.576σ 99% 1% 1 / 100
99.7300204% 0.2699796% 1 / 370.398
3.2906σ 99.9% 0.1% 1 / 1000
99.993666% 0.006334% 1 / 15,788
99.9999426697% 0.0000573303% 1 / 1,744,278
99.9999998027% 0.0000001973% 1 / 506,800,000
99.999 999 999 7440% 0.0000000002560% 1 / 390,600,000,000

## Relationship between standard deviation and mean

.The mean and the standard deviation of a set of data are usually reported together.^ What is the mean, median and mode for this set of data?
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Data sets with a small standard deviation have tightly grouped, precise data.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ The standard deviation is the average distance between the actual data and the mean.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

.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.^ For a certain method, a control has a mean result of 12 with a standard deviation of 2.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ The standard deviation is a statistic calculated from a set of data.
• Standard deviation Information and Courses from MediaLab, Inc. 28 January 2010 0:54 UTC www.medialabinc.net [Source type: FILTERED WITH BAYES]

^ The standard deviation is the average distance between the actual data and the mean.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

.This is because the standard deviation from the mean is smaller than from any other point.^ It is true that the per year average rate of return has a smaller standard deviation for a longer time horizon.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Say we have a reactor with a mean pressure reading of 100 and standard deviation of 7 psig.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ In other words, you would prefer a standard deviation of 12% rather than a 10% return with a deviation of 20%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

The precise statement is the following: suppose x1, ..., xn are real numbers and define the function:
$\sigma(r) = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - r)^2}.$
Using calculus or by completing the square, it is possible to show that σ(r) has a unique minimum at the mean:
$r = \overline{x}.\,$
.The coefficient of variation of a sample is the ratio of the standard deviation to the mean.^ Define a probability distribution by a sequence of control points, then watch how the mean and standard deviation of the distribution change as you move the points.
• Mean and Standard Deviation of a Distribution - Wolfram Demonstrations Project 28 January 2010 0:54 UTC demonstrations.wolfram.com [Source type: FILTERED WITH BAYES]

^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Say we have a reactor with a mean pressure reading of 100 and standard deviation of 7 psig.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

.It is a dimensionless number that can be used to compare the amount of variance between populations with means that are close together.^ We then use the mathematical equations to derive a formal relationship between the input variables and the output variable (in this case the output variable is the fair option premium).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ This model of significance testing is very useful and is often applied to a multitude of data to determine if discrepancies are due to chance or actual differences between compared samples of data.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ What is the 95% confidence interval for the population mean of each model, and what is the 95% confidence interval for the difference between the means of the two populations?
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

.The reason is that if you compare populations with same standard deviations but different means then coefficient of variation will be bigger for the population with the smaller mean.^ Because it is 5.92 for the population standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ No, DC, you are wrong...you found the population standard deviation .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ So, believe it or not, you now know what standard deviation means.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

.Thus in comparing variability of data, coefficient of variation should be used with care and better replaced with another method.^ "The lesson is that one should NOT use the rate of return analysis to compare portfolios of different size.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Statistical methods can be used to determine how reliable and reproducible the temperature measurements are, how much the temperature varies within the data set, what future temperatures of the tank may be, and how confident the engineer can be in the temperature measurements made.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ This model of significance testing is very useful and is often applied to a multitude of data to determine if discrepancies are due to chance or actual differences between compared samples of data.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

## Worked example

.The standard deviation of a discrete random variable is the root-mean-square (RMS) deviation of its values from the mean.^ In fact, it will be reduced by about half the variance (where variance is the standard deviation squared).
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Table 7 displays the group means, standard deviations and the F-values.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

^ The mean scores, standard deviations and the t-values are shown in Table 6.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

If the random variable X takes on N values $extstyle x_1,\dots,x_N$ (which are real numbers) with equal probability, then its standard deviation σ can be calculated as follows:
.
1. Find the mean, $\scriptstyle\overline{x}$, of the values.
2. For each value xi calculate its deviation $\scriptstyle x_i - \overline{x}$ from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations.^ In PerTrac, there are 3 downside deviation calculations, each using a different value for the MAR: 1)Uses a MAR which is defined by the user on the Preferences screen, 2) Uses the Sharpe risk free rate (which can also be defined in Preferences ) as the MAR, and 3) uses zero as the MAR. .
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

^ Table 7 displays the group means, standard deviations and the F-values.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

^ As the r value deviates from either of these values and approaches zero, the points are considered to become less correlated and eventually are uncorrelated.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

This quantity is the variance σ2.
5. Take the square root of the variance.
This calculation is described by the following formula:
$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2},$
where $\scriptstyle \overline{x}$ is the arithmetic mean of the values xi, defined as:
$\overline{x} = \frac{x_1+x_2+\cdots+x_N}{N} = \frac{1}{N}\sum_{i=1}^N x_i.$
If not all values have equal probability, but the probability of value xi equals pi, the standard deviation can be computed by:
$\sigma = \sqrt{\sum_{i=1}^N p_i(x_i - \overline{x})^2}.$
where
$\overline{x} = \sum_{i=1}^N p_i x_i$.
Suppose we wished to find the standard deviation of the distribution placing probabilities 1/4, 1/2, and 1/4 on the points in the sample space 3, 7, and 19.
Step 1: find the probability-weighted mean
3 / 4 + 7 / 2 + 19 / 4 = 9.
Step 2: find the deviation of each value in the sample space from the mean,
\begin{align} 3 - 9 & = -6 \ 7 - 9 & = -2 \ 19 - 9 & = 10. \end{align}
Step 3: square each of the deviations, which amplifies large deviations and makes negative values positive,
\begin{align} (-6)^2 & = 36 \ (-2)^2 & = 4 \ 10^2 & = 100. \end{align}
Step 4: find the probability-weighted mean of the squared deviations,
36 / 4 + 4 / 2 + 100 / 4 = 36.
Step 5: take the positive square root of the quotient (converting squared units back to regular units),
$\sqrt{36} = 6.\,$
.So, the standard deviation of the set is 6. This example also shows that, in general, the standard deviation is different from the mean absolute deviation (which is 5 in this example).^ Say we have a reactor with a mean pressure reading of 100 and standard deviation of 7 psig.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ As sample size increases, the standard deviation of the mean decrease while the standard deviation, σ does not change appreciably.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Motives for teaching: percentage, means, standard deviation and rank order .
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

## Rapid calculation methods

.The following two formulas can represent a running (continuous) standard deviation.^ That is the definition for two standard deviations.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Standard Deviation follows up the band's first release from 1998, Frames of Mind.
• Net Dot Music, Inc. * Electrum - Standard Deviation * Reviews 28 January 2010 0:54 UTC www.netdotmusic.com [Source type: General]

^ I request you for find out the Standard deviation of the following numbers: .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

A set of three power sums s0,1,2 are each computed over a set of N values of x, denoted as xk.
$\ s_j=\sum_{k=1}^N{x_k^j}.$
Note that s0 raises x to the zero power, and since x0 is always 1, s0 evaluates to N.
.Given the results of these three running summations, the values s0,1,2 can be used at any time to compute the current value of the running standard deviation.^ Put into terms for securities, the average return might be 10% with a standard deviation (68% of the time) of 21%.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ It is true that the per year average rate of return has a smaller standard deviation for a longer time horizon.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

^ Let's say we consider minus three standard deviations to be a big loss: the S&P 500 experienced a daily loss of minus three standard deviations about -3.4% of the time.
• Standard Deviation and Monte Carlo 28 January 2010 0:54 UTC efmoody.com [Source type: FILTERED WITH BAYES]

This definition for sj can represent the two different phases (summation computation sj, and σ calculation).
$\sigma= \frac{1}{s_0}\sqrt{s_0s_2-s_1^2}$
Similarly for sample standard deviation,
$s = \sqrt{\frac{s_0s_2-s_1^2}{s_0(s_0-1)}}.$
In a computer implementation, as the three sj sums become large, we need to consider round-off error, arithmetic overflow, and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors:
$A_0=0\,$
$A_i=A_{i-1}+\frac{x_i-A_{i-1}}{i}$
where A is the mean value.
$Q_0=0\,$
$Q_i=Q_{i-1}+\frac{i-1}{i} (x_i-A_{i-1})^2\,$
or
$Q_i=Q_{i-1}+ (x_i-A_{i-1})(x_i-A_i)\,$
sample variance:
$s^2_n=\frac{Q_n}{n-1}$
standard variance
$\sigma^2_n=\frac{Q_n}{n}.$

### Weighted calculation

When the values xi are weighted with unequal weights wi, the power sums s0,1,2 are each computed as:
$\ s_j=\sum_{k=1}^N{w_k x_k^j}.\,$
.And the standard deviation equations remain unchanged.^ Calculate the standard deviation: Using the equation shown above, .
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Equation (3.1) below is another common method for calculating sample standard deviation, although it is an bias estimate.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ When calculated standard deviation values associated with weighted averages, equation (4) below should be used.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

Note that s0 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:
$W_0 = 0\,$
$W_i = W_{i-1} + w_i\,$
and places where 1/i is used above must be replaced by wi/Wi:
$A_0 = 0\,$
$A_i = A_{i-1}+\frac{w_i}{W_i}(x_i-A_{i-1})\,$
$Q_0 = 0\,$
$Q_i =Q _{i-1} + \frac{w_i W_{i-1}}{W_i}(x_i-A_{i-1})^2 = Q_{i-1}+w_i(x_i-A_{i-1})(x_i-A_i)\,$
In the final division,
$\sigma^2_n=\frac{Q_n}{W_n}\,$
and
$s^2_n = \frac{n'}{n'-1}\sigma^2_n\,$
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 1.

## Combining standard deviations

### Population-based statistics

Standard deviations of non-overlapping sub-populations can be aggregated as follows if the size (actual or relative to one another) and means of each are known:
$\mu_{X\cup Y} = \frac{N_X \mu _X + N_Y \mu _Y}{N_X+N_Y}\,\!$
and
$\sigma_{X\cup Y} = \sqrt{\frac{N_X(\sigma_X^2+\mu _X^2) + N_Y(\sigma_Y^2+\mu _Y^2)}{N_X+N_Y} - \mu_{X\cup Y}^2}\,$
where
$X \cap Y \equiv \emptyset. \,$
.For example, suppose it is known that the average American man has a mean height of 70 inches with a standard deviation of 3 inches and that the average American woman has a mean height of 65 inches with a standard deviation of 2 inches.^ The standard deviation is the average distance between the actual data and the mean.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ Say we have a reactor with a mean pressure reading of 100 and standard deviation of 7 psig.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

^ As sample size increases, the standard deviation of the mean decrease while the standard deviation, σ does not change appreciably.
• Basic statistics: mean, median, average, standard deviation, z-scores, and p-value - ControlsWiki 28 January 2010 0:54 UTC controls.engin.umich.edu [Source type: Academic]

The mean and standard deviation for American adults could be calculated as:
$\mu_ ext{height} \approx \frac{50% imes 70 ext{ inches} + 50% imes 65 ext{ inches}}{100%} = \frac{70+65}{2} ext{ inches} = 67.5 ext{ inches}\,$
\begin{align} \sigma_ ext{height} & \approx \sqrt{ \frac{(3^2+70^2)+(2^2+65^2)}{2} -67.5^2 } ext{ inches} \ & = \sqrt{12.75} ext{ inches} \approx 3.5707 ext{ inches} \end{align}
For the more general M non-overlapping data sets X1 through XM:
$\mu_{\{X1 \cup \cdots \cup XM\}} = \frac{ \sum_{i=1}^M { N_{Xi} \mu_{Xi}}}{\sum_{i=1}^M { N_{Xi}}}\,\!$
and
$\sigma_{\{X1 \cup \cdots \cup XM\}} = \sqrt{\frac{\sum_{i=1}^M { N_{Xi} (\sigma_{Xi}^2 + \mu_{Xi}^2)}}{\sum_{i=1}^M {N_{Xi}}} - \mu_{\{X1 \cup \cdots \cup XM\}}^2 }\,\!$
where
$Xi \cap Xj \equiv \emptyset\,\!$
$\forall \,\, i eq j\,\!$
If the size (actual or relative to one another), mean, and standard deviation of two overlapping populations are known for the populations as well as their intersection, then the standard deviation of the overall population can still be calculated as follows:
$\mu_{X\cup Y} = \frac{N_X \mu _X + N_Y \mu _Y - N_{X\cap Y}\mu_{X\cap Y}}{N_X+N_Y-N_{X\cap Y}}\,\!$
and
$\sigma_{X\cup Y} = \sqrt{\frac{N_X(\sigma_X^2+\mu _X^2) + N_Y(\sigma_Y^2+\mu _Y^2) - N_{X\cap Y}(\sigma_{X\cap Y}^2+\mu _{X\cap Y}^2)}{N_X+N_Y-N_{X\cap Y}} - \mu_{X\cup Y}^2} \,\!$
.
If two or more sets of data are being added in a pairwise fashion, the standard deviation can be calculated if the covariance between the each pair of data sets is known.
^ How to calculate standard deviation .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ Okay, now how do I understand the numbers once I have calculated the standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ Table 1 sets out the percentages of responses for the five response categories as well as the means, standard deviations and the rank order based on the mean scores of the items.
• Motives for teaching: percentage, means, standard deviation and rank order 28 January 2010 0:54 UTC www.aare.edu.au [Source type: Academic]

$\sigma_{X1+...+XM} = \sqrt{\sum_{i=1}^M(\sigma_{Xi}^2)-\sum_{i=1}^M\sum_{j=1}^M\operatorname{Cov}(Xi,Xj)} \,\!$
For the special case where no correlation exists between all pairs of data sets, then the relation reduces to:
$\sigma_{X1+...+XM} = \sqrt{\sum_{i=1}^M(\sigma_{Xi}^2)} \,\!$
where
$\operatorname{Cov}(Xi,Xj) = 0 \,\!$
$\forall \,\, \{i,j\}\,\!$

### Sample-based statistics

Standard deviations of non-overlapping sub-samples can be aggregated as follows if the actual size and means of each are known:
$\mu_{X\cup Y} = \frac{N_X \mu _X + N_Y \mu _Y}{N_X+N_Y}\,\!$
and:
$\sigma_{X\cup Y} = \sqrt{\frac{(N_X-1) \sigma_X^2+N_X \mu _X^2 + (N_Y-1) \sigma_Y^2+N_Y \mu _Y^2 - (N_X+N_Y) \mu_{X\cup Y}^2}{N_X+N_Y-1} }\,\!$
where
$X \cap Y \equiv \emptyset. \,\!$
For the more general M non-overlapping data sets X1 through XM:
$\mu_{\{X1 \cup \dots \cup XM\}} = \frac{ \sum_{i=1}^M { N_{Xi} \mu_{Xi}}}{\sum_{i=1}^M { N_{Xi}}}\,\!$
and:
$\sigma_{\{X1 \cup \dots \cup XM\}} = \sqrt{\frac{\sum_{i=1}^M { ((N_{Xi}-1) \sigma_{Xi}^2 + N_{Xi} \mu_{Xi}^2) - (\sum_{i=1}^M {N_{Xi}})\mu_{\{X1 \cup \dots \cup XM\}}^2}}{\sum_{i=1}^M {N_{Xi}-1}} }\,\!$
where
$Xi \cap Xj \equiv \emptyset\,\!$
$\forall \,\, i eq j.\,\!$
.If the size, mean, and standard deviation of two overlapping samples are known for the samples as well as their intersection, then the standard deviation of the samples can still be calculated.^ How to calculate standard deviation .
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ Okay, now how do I understand the numbers once I have calculated the standard deviation.
• How To Calculate Standard Deviation 28 January 2010 0:54 UTC hubpages.com [Source type: FILTERED WITH BAYES]

^ If there are fewer than three data points, or the sample standard deviation is zero, Skewness returns the N/A error value.
• Statistics Used In PerTrac 28 January 2010 0:54 UTC support.pertrac.com [Source type: Academic]

In general:
$\mu_{X\cup Y} = \frac{N_X \mu _X + N_Y \mu _Y - N_{X\cap Y}\mu_{X\cap Y}}{N_X+N_Y-N_{X\cap Y}}\,\!$
and:
$\sigma_{X\cup Y} = \sqrt{\frac{(N_X-1)\sigma_X^2+N_X\mu _X^2 + (N_Y-1)\sigma_Y^2+N_Y\mu _Y^2 - (N_{X\cap Y}-1)\sigma_{X\cap Y}^2-N_{X\cap Y}\mu _{X\cap Y}^2 - (N_X+N_Y-N_{X\cap Y})\mu_{X\cup Y}^2}{N_X+N_Y-N_{X\cap Y}-1} }.\,\!$

## References

1. ^ Dodge, Yadolah (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.
2. ^ Pearson, Karl (1894). "On the dissection of asymmetrical frequency curves". Phil. Trans. Roy. Soc. London, Series A 185: 719–810.
3. ^
4. ^ DasGupta & Haff (2006), "Asymptotic expansions for correlations between different measures of spread". Journal of Statistical Planning and Inference. Vol. 136, pp. 2197–2213
5. ^ Ghahramani, Saeed (2000). Fundamentals of Probability (2nd Edition). Prentice Hall: New Jersey. p. 438.

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

.A measure of variance - actually the SD is the square root of the variance.^ The standard deviation is the square root of the variance.  .
• Statistics in C# : variance, standard deviation, covariance & pearson 28 January 2010 0:54 UTC www.noviway.com [Source type: Academic]

^ The square root of the variance .

^ The standard deviation is the square root of the variance ...
• Article: Standard deviation: a risky measurement tool: standard deviation measures the volatility of a mutual fund, but is imperfect as a risk measurement tool. - Money Digest | HighBeam Research - FREE trial 28 January 2010 0:54 UTC www.highbeam.com [Source type: Academic]

# Simple English

Standard deviation is a concept in statistics that tells you how spread out a set of values is. It can be calculated by considering how far away each value is from the average of all the values.

Standard deviation can be used to measure how consistent or how precise a set of data is.

## Method

In a set of values, you find the standard deviation by following these steps:

1. Find the average of all the values.
2. Subtract the average from each value, giving you their deviations.
3. Square the deviation for each value.
4. Find the average of all these squared deviations.
5. Find the square root of that average.

### Example

We can find the standard deviation of the numbers 3, 7, 7 and 19 as follows.

Step 1: find the average of 3, 7, 7, and 19:

$\left(3+7+7+19\right)/4=9.$

Step 2: find the deviation of each number from the average:

$3 - 9 = -6$
$7 - 9 = -2$
$7 - 9 = -2$
$19 - 9 = 10.$

Step 3: square each of the deviations:

$\left(-6\right)^2=36$
$\left(-2\right)^2=4$
$\left(-2\right)^2=4$
$10^2=100.$

Step 4: find the mean of those squared deviations:

$\left(36+4+4+100\right)/4=36.$

Step 5: find the square root:

$\sqrt\left\{36\right\} = 6.$

So, the standard deviation is 6.

# Citable sentences

Up to date as of December 10, 2010

Here are sentences from other pages on Standard deviation, which are similar to those in the above article.