Standard deviation in statistics

Since 1976 it has been recommended by IUPAC that the notion coefficient of variation should no longer be used instead of relative standard deviation. In statistics it is still known as a feature of a distribution derived from the moments. 

Standard Deviation In statistical analysis, a value equal to the square of the variance. 

Mathematica hasthisfunctionandmanyothersbuiltintoitssetof "add-on" packagesthatare standardwiththesoftware.Tousethemweloadthepackage "Statistics NormalDistribution The syntax for these functions is straightforward we specify the mean and the standard deviation in the normal distribution, and then we use this in the probability distribution function (PDF) along with the variable to be so distributed. The rest of the code is self-evident. 

Experience has shown that correlations of good precision are those for which SD/RMS. 1, where SD is the root mean square of the deviations and RMS is the root mean square of the data Pfs. SD is a measure equal to, or approaching in the limit, the standard deviation in parameter predetermined statistics, where a large number of data points determine a small number of parameters. In a few series, RMS is so small that even though SD appears acceptable, / values do exceed. 1. Such sets are of little significance pro or con. Evidence has been presented (2p) that this simple / measure of statistical precision is more trustworthy in measuring the precision of structure-reactivity correlations than is the more conventional correlation coefficient. 

When specifying atomic coordinates, interatomic distances etc., the corresponding standard deviations should also be given, which serve to express the precision of their experimental determination. The commonly used notation, such as d = 235.1(4) pm states a standard deviation of 4 units for the last digit, i.e. the standard deviation in this case amounts to 0.4 pm. Standard deviation is a term in statistics. When a standard deviation a is linked to some value, the probability of the true value being within the limits 0 of the stated value is 68.3 %. The probability of being within 2cj is 95.4 %, and within 3ct is 99.7 %. The standard deviation gives no reliable information about the trueness of a value, because it only takes into account statistical errors, and not systematic errors. 

The sample standard deviation, s, provides an estimate of the population standard deviation, a. The (n — 1) term in equations (6.4) and (6.6) is often described as the number of degrees of freedom (frequently represented in statistical tables by the parameter v (Greek letter, pronounced nu ). It is important for judging the reliability of estimates of statistics, such as the standard deviation. In general, the number of degrees of freedom is the number of data points (n) less the number of parameters already estimated from the data. In the case of the sample standard deviation, for example, v = n — 1 since the mean (which is used in the calculation of s) has already been estimated from the same data. 

The f-test is similar to the t-test, but is used to determine whether two different standard deviations are statistically different. In the context of chemometrics, the f-test is often used to compare distributions in regression model errors in order to assess whether one model is significantly different than another. The f-statistic is simply the ratio of the squares of two standard deviations obtained from two different distributions ... 

It was assumed that the experimental uncertainties followed Gaussian statistics with equal standard deviation a for all points. Then the standard deviation was determined as ct = (x /v), where v is the number of degrees of freedom in the fit. v is equal to the number of the experimental points less the number of parameters used in the minimization. The best fit with the Frumkin prediction has one free parameter (j6) and gives p =- 2.051 and CT = 0.41 mN/m. The standard deviation in surface tension is small, indicating that the fit with the Frumkin prediction is statistically significant. Similar best fits are obtained for the other surfactants of the homologue series of... 

Values of AHe° and AV = (AHe° - AGe°)/0 (0 = 298.15°K) obtained by Equation 19 had statistical uncertainties (26, 27) that were typically 300-500 cal and 1-2 cal/°K, respectively. Standard deviations in the pK-temperature correlation as analyzed according to Equation 19 with terms involving ACp>0° and higher order terms did not decrease significantly from the two-parameter values of the standard deviations of between 0.01 and 0.04. More accurate pK... 

The results of these interlaboratory studies are reported in USEPA Method Validation Studies 14 through 24 (14). The data were reduced to four statistical relationships related to the overall study 1, multilaboratory mean recovery for each sample 2, accuracy expressed as relative error or bias 3, multilaboratory standard deviation of the spike recovery for each sample and 4, multilaboratory relative standard deviation. In addition, single-analyst standard deviation and relative standard deviation were calculated. 

In a spreadsheet, enter the mean in cell A2 and the standard deviation in cell B2. To find the area under the Gaussian curve from — oo to 900 h in cell C4, we select cell C4 and go to the INSERT menu and choose FUNCTION. In the window that appears, select the Statistical functions and find NORMDIST from the list of possibilities. Double click on NORMDIST and another window appears asking for four values that will be used by NORMDIST. (If you click on help, you will find a cryptic explanation of how to use NORMDIST.)... 

Events following Poisson statistics have the useful property that the standard deviation is equal to VF1/2 this allows the standard deviation in... 

Figure 11. 3H-Thymidine uptake in PBL from S. typhimurium-r/j/ec/ec/ and uninfected calves stimulated with various antigens. Each bar shows the mean of log values of Sist with standard deviations indicated. Statistical analyses were made using Student s t test. (Reproduced with permission from Ref 59. Copyright 1982. Infect. ImmunJ...

A statistical analysis of light-scattering data can compensate for polydispersity. In cumulant analysis, lng(z) is expanded in a power series and coefficients of the different terms are evaluated against the experimentally obtained t, in search of the closest-fitting average selected by the smallness of the standard deviation. In a histogram method, the experimental t is... 

Electrochemical noise can be characterized by some common statistical parameters including the mean, the variance, and the standard deviation. In particular, the standard deviation, o, is used as a measure of the amplitude of the variation in the noise signal. Skew and kurtosis sometimes give indications of the form of corrosion occurring (140). For unfiltered digitized noise data in a time record, the noise resistance, Rn, is... 

Like many data, emission and exposure data are presented as constant values, often a mean with standard deviation. In environmental risk assessment, however, awareness is growing that a stochastic or probabilistic approach is more suitable to obtain insight in the possible risk of chemicals. This also requires expressing exposure data as statistical, probabilistic distributions. Also in this case, the focus should be extended to mixtures. 

Variances, rather than standard deviations, are statistically additive. Thus, provided the individual effects are independent of each other, in Equation (24-3), can be written... 

Historically, risk assessment for noncancer endpoints has been based on the identification of a no observed adverse effect level (NOAEL) from a toxicity study with an animal model. The NOAEL is then divided by appropriate uncertainty factors to take potential inter- and intraspecies differences in response into account. However, this approach does not take into account the size of the toxicity study or the shape of the dose-response curve. The benchmark dose (BMD) approach has been suggested as an alternative to a NOAEL (Crump 1984). A BMD is a dose or concentration that produces a predetermined change (e.g., 10% or 1 standard deviation) in response rate of an adverse effect (called the benchmark response or BMR). A BMDL is the statistical lower confidence limit on the dose or concentration at the BMD. The BMD and BMDL are calculated using mathematical dose-response models, which make appropriate use of sample size and the shape of the dose-response curve (EPA 2009b, 2000a). The BMDL is like a NOAEL (i.e., as a point of departure) and is divided by an appropriate composite uncertainty factor to derive a reference value. 

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