Arithmetic mean standard error

Use individual data points for each experimental concentration (presented as the arithmetic mean standard error of the mean) to construct concentration-response toxicity curves. Such curves are used to calculate midpoint toxicities or NR50 values, as determined by linear regression analysis. Use this value for comparison and ranking against other chemicals, or for relative sensitivity determination of different cell types. 

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. 

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... 

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). 

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. 

This descriptive statistic will be presented by CYP 2C19 metabolizer-status and where applicable by gender, including mean (arithmetic and/or geometric), standard deviation (usual and/or dispersion factor), standard error of the mean, coefficient of variation (in %), median, minimum, maximum, number of observations. 

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. 

Arithmetic mean and standard deviation of two determinations unless otherwise indicated. Relative error from the Karl Fischer value. 

Figure 22.1 Graphical illustration of the data from Table 22.1. To illustrate accuracy and precision the standard deviations have been calculated using formula 22.8. Note the differences between the arithmetic means indicated on the graph by a short vertical bar and the corresponding median values, which are arrowed. For the results from chemists 3 and 4, the difference is fairly large. Chemist 1 has committed very probably a systematic error. On the right, a classic illustration of the precision and accuracy depicted with the aid of a target. This image is less simple than it would appear because there remains an uncertainty in both x and y.

The usual estimate of central tendency is the arithmetic mean. Extensive tabulations of mean bond lengths are available ([5, 6], see also Appendix A to this volume). These are useful for model building, and have a great deal of inherent interest as well. It is therefore worth considering how a mean molecular dimension may best be estimated. Taylor and Kennard [20], using standard statistical methodology [21], have shown that the best estimate of a mean dimension depends crucially on the relative importance of experimental errors and crystal-field environmental effects. 

Notes Data in milliliters thyroid volume in selected regions of the Czech Republic (1991-2006) women (A/ = 2651) data are given in arithmetic means and standard error (se) of means. 

Figure 90.4 loduria levels in schoolchildren before and after the introduction of obligatory iodine prophylaxis in Poland, loduria levels in the same population as in Figure 90.3. The middle point reflects the arithmetic mean, the frame expresses the standard error of mean, and the horizontal lines express standard deviation. Adapted from Zygmunt et al., (2001) with kind permission from Endokrynologia Polska.

Calculate the following descriptive statistics for the data on water hardness (mmoll ) given as follows arithmetic mean, median, standard deviation, variance, standard error, confidence interval at a significance level of 0.01, range, and the interquartile distance - 8.02 7.84 7.98 7.95 8.01 8.07 7.89. 

Standard Error of the Mean A measure of the variability of the distribution of sample arithmetic means with respect to the theoretical population standard deviation. 

The measurements of both surface tension and wetting angle were carried out at a constant temperature. The arithmetic mean of at least three measurements was taken for each experiment. The measure of error was taken as the standard deviation of the arithmetic mean, taking into account the Student s t distribution for confidence level 0.90. 

Figure 21.1—Graphical illustration of the data presented in Table 21.1.To illustrate accuracy and precision, the standard deviations are calculated according to equation (21.8). Variations are indicated on the graph for arithmetic averages by a vertical bar and for the corresponding median values by arrows. For the results from chemist 3, the deviation between the mean and median is large. Chemist 1 has probably committed a systematic error. On the right is the classical target illustration of precision and accuracy. This image appears simpler than it is because there are uncertainties in x and y.

The smaller the random errors of the individual measurements and the larger the number of measurements, n, the smaller the deviation of the arithmetic average from the true mean. The empirical sample standard deviation, s, is a measure of the measurement error [Eq. (3)]. 

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