Root mean square error definition

Maximum absolute deviations (MAD) and root mean square errors (RMSE) Definition of the basis sets 6-31G(d,p) (basis I), 6-311 +G(d,p) (basis n), Experimental values were obtained using the data in Ref. [48]... 

By definition, the standard deviatirai is the root-mean-square deviation about the mean value. It does not provide an indicator of the statistical error about the mean of multiple measurements. If the distribution is unimodal and not too skewed, then the standard deviation will give a reasonable indication of dispersity.. 

Unfortunately, definitions, nomenclature, and abbreviations used for performance criteria are sometimes confusing (Frank and Todeschini 1994 Kramer 1998). For instance in the abbreviations MSEC, PRESS, RMSEP, SEC, SEE, SEP, E means error or estimate, R means residual or root, and S means squared or standard or sum. To make it not too complicated, at least in these examples, C is always calibration, M is mean, and P is prediction. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

A more rigorous definition of uncertainty (Type A) relies on the statistical notion of confidence intervals and the Central Limit Theorem. The confidence interval is based on the calculation of the standard error of the mean, Sx, which is derived from a random sample of the population. The entire population has a mean /x and a variance a. A sample with a random distribution has a sample mean and a sample standard deviation of x and s, respectively. The Central Limit Theorem holds that the standard error of the mean equals the sample standard deviation divided by the square root of the number of samples ... 

Measured accurate masses, when used to assign molecular formulas, should always be accompanied by their mass accuracies. [52], Ideally, this can be done by giving the mean mass value and the corresponding error in terms of standard deviation through several repeated measurements of the same ion. [40] This is definitely not identical to the error usually provided with a mass spectrometer s software, where the error is based on the difference of a single pair of calculated and measured values. The reduction of the average mass error goes with the square root of the number of determinations (Chap. 3.5.4) [39]. 

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