Standard error of a mean

What is the standard error of a mean Why is the standard deviation of the mean lower than the standard deviation of the data points in a set ... 

The standard error of a mean based on 2 samples (4 carrots per sample), 2 aliquots per sample, and 2 runs per sample, or 8 observations, was 2.8 for falcarinol and 4.8 for falcarindiol. For single observations, the respective standard deviations were 15.2 and 24.8. About 70% of the variation for falcarinol was associated with sample, but only about 25% for falcarindiol. For falcarindiol, variation associated with the assay differences contributed most of the variation. Precision for both toxicants... 

The standard error of a point estimator is an average error, where the average is with respect to repeated experiments of the same type. More precisely, it is the standard deviation of the point estimator. TTiis subsection begins with a basic result for estimating the standard error of a mean or probability estimate and then illustrates it using the examples. ... 

Variance of mean Standard error of a mean Coefficient of variation Per cent variability... 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

The estimate of variability for a sample mean is the standard error of the mean ... 

The mean of several readings (x) will make a more reliable estimate of the true mean (yu) than is given by one observation. The greater the number of measurements (n), the closer will the sample average approach the true mean. The standard error of the mean sx is given by ... 

Figure 4.1 Time-course of free-radical production during aerobic (a) or anoxic (b) reperfusion of the isolated rat heart. Radical production was assessed using e.s.r. and quantified as the formation of a Af-tert-butyl-a-phenylnitrone (PBN) spin adduct. After a 35 min stabilization period of aerobic perfusion, hearts were made globally ischaemic for 15 min. Hearts were then reperfused, either with oxygenated buffer (a) (n = 6), or with anoxic buffer, switching to an oxygenated buffer after 10 min (b) (n = 5). The bars represent the standard errors of the means. Redrawn with permission from Garlick et af. (1987).

Table 4.1 Effect of selected thiols, disulphides, amino acids and antioxidants on the time to the onset and the time to reach maximal ischaemic contracture in isolated perfused rat hearts. Hearts were perfused for a control period of 10 min at the end of which global low-flow (10% of control) ischaemia was initiated. The interventions described above were included in the perfusion fluid 5 min prior to the onset and throughout the ischaemic period. The data are shown as means standard errors of the means (n = 6)...

Fig. 4 Mean cumulative urinary excretion of nitrofurantoin after oral administration of a 100-mg macrocrystalline capsule of fasting (O) and nonfasting ( ) subjects and a 100-mg microcrystalline tablet to fasting ( ) and nonfasting ( ) subjects. Vertical bars represent standard errors of the mean. (From Ref. 7.).

Figure 3 Migration inhibition assessment of ECRF24 and MDA-MB-231 cells after exposure to compound 1 and 3. Wound closure in ECRF24 cultures after 7 h of incubation with concentration ranges of 1 (A) and 3 (B). (C) Typical images of the wound at the beginning of the experiment (culture medium as a control) and after 7 h of incubation with 3, 60 pM. Error bars represent standard error of the mean. P < 0.05.

The uncertainty in the finite-data free energy values, 5 A A, can be estimated as twice the standard error of the mean, 8AA = 2[Pg.240]

The ratios of the experimental to the calculated values for deposition efficiency for each generation are shown for each of the experiments in Figures 2-4. The error bars represent the standard error derived from the mean measured deposition fraction for the generation. A coefficient of variation was calculated for the measured deposition in each generation which was used to obtain an estimate of the standard error of the ratio. It was assumed that no additional variability in the ratio was introduced by the calculated deposition fraction. The mean ratio for all six sets of cast measurements is shown in Figure 5. The error bars in Figure 5 represent the standard error of the mean of the six experiments. 

Fig. 4.6. The integrated signals of harmonics from an H2 gas (lO.OTorr), a D2 gas (lO.OTorr), and a gas of their mixture (lO.OTorr) measured as a function of the harmonic order q. The signals are normalized to those from the H2 gas and vertical errors represent SEM (standard error of the mean) for 800 laser shots...

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... 

The standard deviation of the collection of all possible means of samples of a given size, called the standard error of the mean, depends on both the standard deviation of the original population and the size of the sample. 

Table 9.21 gives the yields, times to maximum yield, retention times and least detectable amounts of the herbicide esters or ethers prepared using the above method. In no instance was the standard error of the mean yield >2%. The least detectable amount is based on a peak giving a response of twice the background signal. 

Standard error of the mean (SEMI Is a statistical assessment of the increased confidence in a result ighen ft isthe meert of a iiQfinM numw or. replicates rather than a single measurement... 

Whereas precision (Section 6.5) measures the reproducibility of data from replicate analyses, the accuracy (Section 6.4) of a test estimates how accurate the data are, that is, how close the data would represent probable true values or how accurate the analytical procedure is to giving results that may be close to true values. Precision and accuracy are both measured on one or more samples selected at random for analysis from a given batch of samples. The precision of analysis is usually determined by running duplicate or replicate tests on one of the samples in a given batch of samples. It is expressed statistically as standard deviation, relative standard deviation (RSD), coefficient of variance (CV), standard error of the mean (M), and relative percent difference (RPD). 

Additional measurements were made with the 17-)im sizing screen to obtain more information on the variability of our measurement techniques. Eight lint samples from a single source of cotton were analyzed by the procedures outlined previously. The dust levels obtained in this test were 11.7, 12.1, 13.5, 11.8, 10.8, 11.2, 10.9, and 9.7 mg, respectively, per 20 g of lint. The mean and standard deviation of these measurements were 11.5 and 1.1, respectively. The estimated standard error of the mean was 0.42, and the interval from 10.5 to 12.5 represented a 95% confidence interval for the lot mean. 

This example shows that the standard deviation of the sampling distribution is less than that of the population. In fact, this reduction in the variability is related to the sample size used to calculate the sample means. For example, if we repeat the sampling experiment, but this time based on 15 rather than 10 random samples, the resulting standard deviation of the sampling is 0.159, and on 25 random samples it is 0.081. The precise relationship between the population standard deviation a and the standard error of the mean is ... 

The area under the entire curve equals a probability of unity. Over the interval from — lo-to +I0-, the integrated area equals 68.3% of the total area under the curve thus, the probability of finding a value within that interval is 0.683. For la, the probability rises to 0.955, and for 3o-, the probability is 0.997. Note that for a set of Ndata values, the estimated mean will progressively deviate less and less from the true mean as the number of data points increases. This so-called standard error of the mean (v) is inversely related to N. See Statistics A Primer)... 

Note the role played by the standard error in the formula for the confidence interval. We have previously seen that the standard error of the mean provides an indirect measure of the precision with which we have calculated the mean. The confidence interval has now translated the numerical value for the standard error into something useful in terms of being able to make a statement about where jl lies. A large standard error will lead to a wide confidence interval reflecting the imprecision and resulting poor information about the value of jjl. In contrast a... 

Figure 2. Distribution of mean concentrations with wind direction for 4 crustal elements measured on a streaker sampler at ground level during one week in June 1978. Radial bars indicate I standard error of the mean. The numbers at the end of the bars indicate the number of 2-h samples from each direction.

A meta-analysis for continuous data cannot be calculated unless the pertinent standard deviations are known. Unfortunately, clinical reports often give the sample size and mean ratings for the various groups but do not report the standard deviations (or standard error of the mean), which are necessary for effect size calculations. Thus, investigators should always report the indices of variability (e.g., confidence intervals, SDs) for the critical variables related to their primary hypothesis. 

When we are dealing with samples rather than populations, we cannot use the standard normal deviate, Z, to make predictions since this requires knowledge of the population mean and variance or standard deviation. In general, we do not know the value of these parameters. However, provided the sample is a random one, its mean 5 is a reliable estimate of the population mean p, and we can use the central limit theorem to provide an estimate of o. This esti mate, known as the standard error of the mean, is given by ... 

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