Sampling from populations - the SEM

Triplicate aliquots were taken for particle size analysis and two of those aliquots were mixed for BET surface area analysis results are in Table III. The nine samples were individually sieved for size distribution. A chi-squared test was performed on each triplicate set in order to check the apparent efficiency of composite mixing. For all three composite samples, there was a 90 percent probability that each of the three replicates from each composite sample came from the same population. The A and C samples were combined and evaluated for surface area by nitrogen adsorption (BET). The B samples were then subjected to scanning electron microscopy (SEM) analysis. 

The SEM describes the degree of uncertainty present in the assessment of the population mean on the basis of the sample mean. This degree of uncertainty is due to sampling error. Conversely, it facilitates statements of the degree of certainty associated with the results obtained from a single sample. 

The SEM is shown in action in Figure 4.3. This contrasts what would happen if we were to take either small or large samples from the same population. Small samples are not very precise and the sample means would be badly spread out. Hence, the SEM would be large. With the larger samples, the means would be more tightly clustered and the SEM smaller. The SEM has thus achieved what we wanted. It has acted as a indicator of likely sampling error - large errors are quite likely with the smaller samples, but big samples are less error prone. 

The obvious question is how wide should the interval be Clearly, the wider the interval is, the greater our confidence that it will include the true population mean. For example, interval (a) in Figure 5.1 only covers a range of about 0.2 mg. Since we know, from the SEM, that a sampling error of 0.3 mg is perfectly credible, we would have very little confidence that the true mean will fall within any such narrowly defined interval. 

In most analytical experiments where replicate measurements are made on the same matrix, it is assumed that the frequency distribution of the random error in the population follows the normal or Gaussian form (these terms are also used interchangeably, though neither is entirely appropriate). In such cases it may be shown readily that if samples of size n are taken from the population, and their means calculated, these means also follow the normal error distribution ( the sampling distribution of the mean ), but with standard deviation sj /n this is referred to as the standard deviation of the mean (sdm), or sometimes standard error of the mean (sem). It is obviously important to ensure that the sdm and the standard deviation s are carefully distinguished when expressing the results of an analysis. 

The Cl for a mean essentially describes where the population mean (u) lies with respect to the sample mean X with a given probability. If several means are available from different groups of measurements of the same sample, the individual means will also be distributed normally around the grand mean. The random variation in a population of means is described by the SEM ... 

Quite a different notion is the distribution of a sample statistic such as the sample mean or the sample standard deviation. For instance samples taken from a population with parameters p and have themselves a distribution with mean p and variance = o /n, where n is the sample size. The square root of is often called the standard error of the mean or SEM ... 

M ABA to four leaves had caused an increase of the fraction of totally closed stomata from on the average 0.2 in the control leaf to 0.4, and a shift of the population of open stomata toward classes with small apertures. From these data, stomatal conductances for water vapor (and COg) were computed (Fig. 5 center histogram). Based on a saturation curve of photosynthesis of an untreated Xanthium leaf, rates of photosynthesis were estimated for each aperture class. Integration yielded total assimilation rates of the control and the ABA-treated leaves (right histogram). The observed increase in the number of closed stomata, in combination with the narrowing of the open stomata, caused in the ABA-treated leaf a computed decrease of the assimilation rate from 18.2 to 10.7 jitmol m s The directly measured rates had been 20.6 and 7.7 jumol m s The agreement is satisfactory in view of the fact that only about 0.2% of the stomatal population had been sampled under the SEM. 

The use of the mean with either the SD or SEM implies, however, that we have reason to believe that the sample of data being summarized are from a population that is at least approximately normally distributed. If this is not the case, then we should rather use a set of statistical descriptions which do not require a normal distribution. These are the median, for location, and the semiquartile distance, for a measure of dispersion. These somewhat less familiar parameters are characterized as follows. 

The standard deviation of a group of sample means taken from the same population (SEM) ... 

Notes-, (a) Based on the U.S. NHANES III (1988-1994) laboratory sample, weighted to national population, (b) Excluding patients with thyroid disease and goiter, (c) GM - Geometric means. Weighted geometric mean standard error of the mean (SEM) and 95% confidence interval, (d) Ul conversion factor from pg/l to nmol/l = x 7.9. 

SEM/EDS analyses of all particles <1 mm differentiated microplastics, including multi-colored plastic spheres, from other materials such as coal ash. Based on dense urban populations adjacent to the lakes that employ combined sewage overflow, and the convergence of lake currents near our sample sites, we believe the microplastic spheres may be microbeads used in consumer product applications, such as those used in facial cleansers. 

---