Error arithmetic mean

HF/6-31G /MP2(fc)/6-31G, respectively, the mean of nine errors for each method is 3.0/2.3/1.9. Errors are given in the Errors column as HF/3-21G< )/HF/6-31G /MP2/6—31G A minus sign means that the calculated value is less than the experimental. The numbers of positive and negative deviations from experiment and the average errors (arithmetic means of the absolute values of the errors) are summarized at the bottom of the Errors column. Calculations are by the author, references to experimental measurements are given for each measurement. Some molecules have calculated minima at other dihedrals in addition to those given here, e.g. FCH2CH2F at 180°. Errors are presented HF/3-21G( VHF/6-31G /MP2/6-31G. a[lg], pp.151, 152... 

The measurement error might be described by the mean error (arithmetic mean value), the variance of the error, but also by the mean square error, the mean error power, and the root mean square error. 

When we report the result of a measurement a , there are two things a person reading the report wants to know the magnitude (size) of the measurement and the reliability of the measurement (its scatter ). If measuring errors are random, as they very frequently are, the magnitude is best expressed as the arithmetic mean p of N repeated tr ials xi... 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

A problem with the overall approach for liquid mixtures is that suitable averages must be used when calculating B, although errors in B are partly offset by the logarithmic term in Eq. (1). It is also necessary to decide at what temperature the properties of air should be evaluated. In [66] it was suggested that Cp a and should be evaluated at the arithmetic mean of the... 

Here the last expression was found by taking the arithmetic mean between the two forms, Eq. 11.45 and Eq. 11.74. Formulas of this type have actually been used in the cellular method" for treating crystals, but our own experience from work on atoms is that the orbital energies sk seem to be the quantities in the HF scheme which are most easily influenced by numerical uncertainties and errors. Even if Eqs. 11.74 and 11.75 are practically simpler to handle than Eq. 11.45, they are probably less numerically reliable. Further investigations on this point are desired. 

Variance The mean square of deviations, or errors, of a set of observations the sum of square deviations, or errors, of individual observations with respect to their arithmetic mean divided by the number of observations less one (degree of freedom) the square of the standard deviation, or standard error. 

T divided by the viscosity of the solvent r s. For n-octane this number is 837 K/cP at T = 323 K. The results of the fitting process are all below this theoretical value. This is not surprising, since even in the case of dilute solutions of unattached linear chains, the theoretical values are never reached (see Sect. 5.1.2). In addition the experimental T/r s values differ considerably for the different labelling conditions and the different partial structure factors. Nevertheless, it is interesting to note that T/r s for the fully labelled stars is within experimental error the arithmetic mean of the corresponding core and shell values. 

One then proceeds to calculate a value of the rate constant for each pair of points separated by a time A [i.e., a value is calculated from the points corresponding to (0 and A), and (A + tj, 2ti and (A + 2tfj, etc.]. The arithmetic mean of these values is a good representative value of the rate constant. In this technique each data point is used once and only once, and the probable errors of the quantities that are averaged are all of the same order of magnitude. For the first-order case it is apparent from equation 3.1.8 that the average value of the rate constant is given by... 

Most commonly, location is described by giving the (arithmetic) mean and dispersion by giving the standard deviation (SD) or the standard error of the mean (SEM). The calculation of the first two of these has already been described. If we again denote the total number of data in a group as N, then the SEM would be calculated as... 

Note that z can be larger than the number of objects, n, if for instance repeated CV or bootstrap has been applied. The bias is the arithmetic mean of the prediction errors and should be near zero however, a systematic error (a nonzero bias) may appear if, for instance, a calibration model is applied to data that have been produced by another instrument. In the case of a normal distribution, about 95% of the prediction errors are within the tolerance interval 2 SEP. The measure SEP and the tolerance interval are given in the units of v, and are therefore most useful for model applications. 

The mean squared error (MSE) is the arithmetic mean of the squared errors,... 

Bias is the total systematic error (there may be more than one component contributing to total systematic error). It is the (positive or negative) difference (A) of the population mean (p, the limiting value of the arithmetic mean for u-a qo) from the (known or assumed) trae value (x). A = p - t. Therefore bias is the lack of traeness. 

The difference between the two means is a factor of 2.2. This value is larger than the expected error of equation (9.7). Thus, a channel with a variation in slope and cross section along its length will have a higher K2 value computed from arithmetic means than an otherwise equivalent channel that does not have variation in slope and cross section. It may not be a coincidence that Moog and Jirka s calibration of Thackston and Krenkel s equation for flumes is an adjustment by a factor of 0.69 to represent held measurements. We need to pay attention to the impact that these variations in natural rivers and streams have on our predictive equations for K2. 

Mathematical analysis of error curves leads to the conclusion that the arithmetic mean x of individual values is the best estimation of the real mean //, (Fig. 21.2). The features and the symmetry of this curve show that ... 

Moreover, covalent bond distances are often related to one another in an additive manner the bond distance A- -B is equal to the arithmetic mean of the distances A—A and B —B. For example, the C—C distance in diamond is 1.542 A and the Cl—Cl distance in Clj is 1.988 A. The arithmetic mean of these, 1.765 A, is identical with the Cl—Cl distance 1.766 0.003 A found in carbon tetrachloride to within the the probable error of the experimental value.5 In consequence, it becomes possible to assign to the elements covalent radii such that the sum of two radii is approximately equal to the equilibrium inter-nuclear distance for the two corresponding atoms connected by a single covalent bond. 

Proportion of individuals for each taxonomic group that contributes to the total individuals in a field site under study. The arithmetic mean (plus or minus the standard error, plus or minus the standard deviation), minimum and maximum for the area are also calculated. Volume 2(4). 

Table 7.1 presents for examination 43 bond lengths and 19 bond angles, taken from 20 molecules. For each of these parameters the deviation from experiment (calculated - experimental value) is shown for B3LYP, M06, TPSS, and MP2 (with the 6-31G basis in each case). The mean absolute deviations from experiment (arithmetic mean of the unsigned errors), MAD, are ... 

Table 7.5 Energy errors for hydrogenation reactions, isomerizations, bond separation reactions, and proton affinities, using four different methods the basis set is 6-31G. The errors, inkJ mol-1, in each case the arithmetic mean of the absolute deviations from experiment of ten reactions, were calculated from the data in Hehre [86]...

The concentration of each latex was determined gravimetrically. Three 1.0 ml samples of each latex were weighed to the nearest 0.1 mg. The samples were placed in a vacuum oven (55° C, -100 kPa) for two hours. The dried samples were allowed to cool to room temperature in a deseccator and weighed again to the nearest 0.1 mg. Weight percent values were calculated for the three samples of each latex. The arithmetic mean of the three weight percent values and the standard error of the means for each latex are listed in the third and fourth columns, respectively, of Table I. 

The quantity s2/n, which represents the variability among arithmetic means (i.e., simple averages) of n repetitions, is known as the standard error of the mean, and is abbreviated either as SEM or SE. Results of assays involving n repetitions, such as the absorbance results reported earlier as x SD = 0.42 0.09 are also frequently reported as x SE, which in this case is 0.42 (0.09/V3) = 0.42 0.05. Confusion can arise if either this x SE or the x SD format is used without specific indication of whether it is SD or SE that follows the reported mean. The latter is preferable whenever the purpose is to represent the precision of the assay s summary result rather than the variability of the individual measurements that have contributed to it. This is almost always the case with chemical analyses, and we recommend the x SE notation, supplemented by the number of replicates, n, for general use. Thus, in the example, we would report an absorbance of 0.42 0.05 (SE), from three replications. 

Assuming two experimenters to have measured the same ratio, their results being Rx and R2, with probable errors of a and b respectively then, instead of adopting the arithmetic mean (R1+R2)/2, it is better to employ the weighted mean, the results being weighted inversely as the squares of their probable errors. The weighted mean is then... 

So far the discussion has dealt with the errors themselves, as if we knew their magnitudes. In actual circumstances we cannot know the errors by which the measurements Xj deviate from the true value Xq, but only the deviations (x,- — x) from the mean T of a given set of measurements. If the random errors follow a Gaussian distribution and the systematic errors are negligible, the best estimate of the true value Aq of an experimentally measured quantity is the arithmetic mean x. If you as an experimenter were able to make a very large (theoretically infinite) number of measurements, you could determine the true mean /jl exactly, and the spread of the data points about this mean would indicate the precision of the observation. Indeed, the probability function for the deviations would be... 

Note that the centroid is the simple arithmetic mean of the vectors of the cluster members, and this mean is frequently used to represent the cluster as a whole. In situations where a mean is not applicable or appropriate, the median can be used to define the cluster medoid (see Kaufman and Rousseeuw for details). The square-error (also called the within-cluster variance), for a cluster is the sum of squared Euclidean distances to the centroid or medoid for all s items in that cluster ... 

In most analyses, replicate measurements are carried out so that random errors can be estimated. In such cases the average value of the analyte level is normally expressed using the arithmetic mean, x, (often simply called the mean) given by Equation 2 ... 

These calculations are not, however, all of the same precision, and a simple arithmetic mean will give a disproportionately high weight to the less reliable points. If estimates can be made of the relative precision for each pair of points, then for completely random errors the calculated values of k should be weighted for averaging according to their expected errors. For example, a value having an estimated error of 2 per cent... 

An analytical result (arithmetical mean of a series of parallel measurements) can only have a bias and random error according to the following dependence [2] ... 

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