Deviation, standard

Standard deviation is a widely used measure of dispersion of data in a given data set about the mean and is expressed by 

The following three properties of the standard deviation are associated with the widely used normal distribution discussed later in the chapter  

Calculate the standard deviation of the data set given in Example 2.1 Using the Example 2.1 data set and the calculated mean value (i.e., m = p = 5.5) in Equation (2.3), we get 

The standard deviation (denoted s or sd) is a measure of patient-to-patient variation. There are other potential measures but this quantity is used as it possesses a number of desirable mathematical properties and appropriately captures the overall amount of variability within the group of patients. 

So if the standard deviation is 3, the variance is 9, if the variance is 25 then the standard deviation is 5. 

The method of calculation of the standard deviation seems, at least at face value, to have some arbitrary elements to it. There are several steps  

People often ask why divide by — 1 rather than Well, the answer is a fairly technical one. It can be shown mathematically that dividing by n gives a quantity that, on average, underestimates the true standard deviation, particularly in small samples and dividing by n — 1 rather than n corrects for this underestimation. Of course for a large sample size it makes very little difference, dividing by 99 is much the same as dividing by 100. 

Another frequent question is why square, average and then square root, why not simply take the average distance of each point from the mean without bothering about the squaring Well you could do this and yes you would end up with 

The standard deviation provides a measure of variability about the mean or average value. If two data sets have the same mean, but their data range differ, so will their standard deviations. The larger the range, the larger the standard deviation. 

For instance, using our previous example, the six data points—10, 13, 19, 9, 11, and 17— have a range of 19-9 = 10. The standard deviation is calculated as 

Given a sample set is normally distributed, the standard deviation has a very useful property, in that one knows where the data points reside. The mean +1 standard deviation encompasses 68% of the data set. The mean 2 standard deviations encompass about 95% of the data. The mean + 3 standard deviations encompass about 99.7% of the data. For a more in-depth discussion of this, see D.S. Paulson, Applied Statistical Designs for the Researcher (Marcel Dekker, 2003, pp. 21-34). 

In this book, we restrict our analyses to data sets that approximate the normal distribution. Fortunately, as sample size increases, even nonnormal populations tend to become normal-like, at least in the distribution of their error terms, e = (yj—y), about the value 0. Formally, this was known as the central limit theorem, which states that in simulation and real-world conditions, the error terms e become more normally distributed as the sample size increases (Paulson, 2003). The error itself, random fluctuation about the predicted value (mean), is usually composed of multiple unknown influences, not just one. 

The reproducibility is typically expressed as the standard deviation of the mean value. At an infinite number of measurements, the distribution of individual results is represented by a Gaussian curve. Therefore, the distance of the inflection points from the true value ft is used as a measiu-e for the scatter. This is denoted as theoretical standard deviation 7. However, in practice, neither an infinite number of measuring values exists nor is the true value of a sample known. The measured distribution of a limited niunber of individual values no longer corresponds to a normal distribution but rather to a t-distribution. At repetitions of a measurement, one obtains s as an estimate for the standard deviation [20]  

As the number of measurements increases, the estimated values x and s approach the true values ft and a, respectively. 

The standard deviation is a measure for random error of an analysis, which depends on the method and the sample composition. The relative random error e serves as a measure for the precision of an analysis the smaller the error, the higher the precision. 

Using the relative random error, an analyst may quickly prove whether the obtained measuring values, independent of their magnitude, are within the precision range of the method. 

The average difference between the mean and a value is called the standard deviation. For example, in the case above, the difference between each measurement (the 

Using the accident data from above, the standard deviation would be calculated as follows  

Therefore, on average, the number of reported accidents at any of the six sites will be 2.16 away from 4.7. Or, the number of accidents reported at each site, on average, will be 4.7 give or take 2.16. This is sometimes written as 4.7 2.16. 

The standard deviation can also be used to determine the expected ranges for scores in a given distribution. By definition, we expect to find approximately 68 percent of the population between +1 and -1 standard deviations, approximately 95 percent of the population between + 2 and - 2 standard deviations, and 99 percent of the population between +3 and -3 standard deviations. A population is the total number of possible sources from which data could be obtained. For example, all of the employees in a company could be defined as the population while those from which data is actually collected is called a sample. A sample is a subset of the total population. 

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In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

The points are within the stated standard deviation and are randomly distributed about the zero axis. 

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

H. The next cards provide estimates of the standard deviations of the experimental data. At least one card is needed with non-zero values. Units are the same as those of the VLE data. FORMAT(4f10.2,I2). ... 

SDY(I) cols 1-10 standard deviation of pressure measurement SDX(I,l)cols 11-20 standard deviation of temperature measurement SDX(l,2)cols 21-30 standard deviation of liquid composition measurement... 

SDZ(I) cols 31-40 standard deviation of vapor composition measurement... 

LOAD VLE DATA, ESTIMATED STANDARD DEVIATIONS, AND INITIAL PARAMETER EST IMATES... 

READ STANDARD DEVIATIONS OF VLE MEASUREMENTS- ND IS THE NUMBER OF STANDARD DEVIATIONS WHICH ARE DUPLICATED (1.UE.NO.LE-NN). 

The method allows variables to be added or multiplied using basic statistical rules, and can be applied to dependent as well as independent variables. If input distributions can be represented by a mean, and standard deviation then the following rules are applicable for independent variables ... 

The mean (m) and the standard deviation (a) of the population of localized AE sources over each cell of the (ATi,AT2) histogram are calculated. 

The threshold value for screening out non significant AE sources is calculated on the basis of the mean (m) and of the standard deviation (a) TH = (m+Na), where N is an input floating variable. 

TEST FOR CONCENTRATED SOURCES THRESHOLD VALUE ON STANDARD DEVIATION (intNa). SCREEN ALL SOURCES WHOSE COUNTS ARB BELOW THRESHOLD... 

Table 2 compares between the VIGRAL results and mechanical measurements of the simulated FBH defects. The table lists the size of the reflecting surface,, its depth location, the Yd, and the standard deviation of the depth information, o>i( y ). We note an excellent agreement between the VIGRAL and the mechanical measurements both in size and depth of the defects. 

BAM produces and distributes calibrating films for the measurement of the standard deviation of the density. 

Figure 3 Feature relevance. The weight parameters for every component in the input vector multiplied with the standard deviation for that component are plotted. This is a measure of the significance of this feature (in this case, the logarithm of the power in a small frequency region.)...

The combined result of two such determinations yielded a leak size figure of 8.8% of the feed flow (with a relative standard deviation of less than 5%). This figure could sufficiently explain the product quality problems experienced, whose alternative explanation in turn was catalyst poisoning. 

Although a seemingly odd mathematical entity, it is not hard to appreciate that a simple one-dimensional realization of the classical P x , t) can be constructed from the familiar Gaussian distribution centred about x by letting the standard deviation (a) go to zero. 

The Heisenberg uncertainty principle offers a rigorous treatment of the qualitative picture sketched above. If several measurements of andfi are made for a system in a particular quantum state, then quantitative uncertainties are provided by standard deviations in tlie corresponding measurements. Denoting these as and a, respectively, it can be shown that... 

Probability theory shows that tire standard deviation of a quantity v can be written as... 

Figure Cl.5.15. Molecular orientational trajectories of five single molecules. Each step in tire trajectory is separated by 300 ms and is obtained from tire fit to tire dipole emission pattern such as is shown in figure Cl.5.14. The radial component is displayed as sin 0 and tire angular variable as (ji. The lighter dots around tire average orientation represent 1 standard deviation. Reprinted witli pennission from Bartko and Dickson 11481. Copyright 1999 American Chemical Society.

The two main ways of data pre-processing are mean-centering and scaling. Mean-centering is a procedure by which one computes the means for each column (variable), and then subtracts them from each element of the column. One can do the same with the rows (i.e., for each object). ScaUng is a a slightly more sophisticated procedure. Let us consider unit-variance scaling. First we calculate the standard deviation of each column, and then we divide each element of the column by the deviation. 

The result is that each column vector has unit standard deviation. 

The model building step deals with the development of mathematical models to relate the optimized set of descriptors with the target property. Two statistical measures indicate the quality of a model, the regression coefficient, r, or its square, r, and the standard deviation, a (see Chapter 9). 

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