Standard deviation of a sample

So how does this help us determine n As we know from our previous discussion of the Central Limit Theorem [2], the standard deviation of a sample from a population decreases from the population standard deviation as n increases. Thus, we can fix fi0 and yua and adjust the a and [3 probabilities by adjusting n and the critical value. 

Generally, it is easier to calculate the standard deviation of a sample by using an equivalent formula, as follows ... 

Recoveries of mercury ranged from 83% at the 1 pg mercury level to 96.1% at the 100 pg mercury level in turf, 94% at the 0.5 pg mercury level to 99.5% at the 4 pg mercury level in cracked whole barley, and 101% at the 0.1 pg mercury level to 94% at the 0.5 pg mercury level in wheat. For five-gram barley samples containing less than 5 pg mercury, the standard deviation of a sample determination was 0.12. 

The standard deviation of a sample (Eq. 20.3) is an estimate of the standard deviation of the corresponding population distribution. 

Usually no value for c is available and one only wants to control whether there is, for example, a 95 % chance that the trae mean value is above a certain specification. Assume that values of thickness measurements are normally distributed. Let fj. be the tme mean thickness of the geomembrane, m the measured mean value and s the standard deviation of a sample of an agreed upon number n of randomly taken specimen. (For example 10 specimens is a reasonable number.) The m- ... 

Coefficient of Variation The ratio of the standard deviation, of a sample or population. It is sometimes defined as the percent e of the ratio as opposed to the fiuction. The coefficient of variation is a measure of relative dispersion. 

Sample Space n The sample space is a set (or list) of all the possible outcomes of a random experiment. The set of all possible samples of an experiment is the sample space for the experiment. The set is referred to as exhaustive since it represents all possible outcomes of the experiment. The sample space is usually denoted by S or fl Sample Standard Deviation n The positive square root of the sample variance. This is equivalent to the standard deviation of a sample of the population. 

Across the entire compound set, the CD detector showed the lowest detection limits. For most compounds, the calculated LOD was the same order of magnitude as the observed. The Chiralyzer showed the next best LOD however, the calculated values were often higher than the observed values. Since the LOD is calculated using the standard deviation of a sample set, the higher LOD values are indicative of scatter. The Chiralyzer, therefore, is considered a sensitive detector but due to the scatter not precise. The lack of precision, howevei does not deter its use of this detector for screening purposes. The PDR Chiral and the ORD are ranked third and fourth in sensitivity, respectively. 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

Equation (3) allows the calculation of the distance traveled axially by a solute band before the radial standard deviation of the sample is numerically equal to the column radius. Consider a sample injected precisely at the center of a 4 mm diameter LC column. Now, radial equilibrium will be achieved when (o), the radial standard deviation of the band, is numerically equal to the radius, i.e., o = 0.2 cm. 

Validation also applies to software. In a simple example, you could create an Excel spreadsheet template with fixed formulae to calculate the mean and standard deviation of a range of data. To validate this template you would enter a set of sample data and verify the template-calculated results against the manually calculated results. In order to be confident that the template could be used for further data sets you would password-protect the cell formulae and verify that they cannot be altered without it. 

The small peak volumes typical of samples eluted from small bore columns and short small diameter particle columns used in high-speed liquid chromatography place severe demands on the dispersion characteristics of all components of the liquid chromatograph. The standard deviation of a peak eluting from a column is given by... 

For a graphical comparison of the correlation [r(Sr)] and the standard deviation of the samples used for calibration (Sr), a value is entered for the SEP (or SEE) for a specified analyte range as indicated through the standard deviation of that range (Sr). The resultant graphic displays the Sr (as the abscissa) versus the r (as the ordinate). From this graphic it can be seen how the correlation coefficient increases with a constant SEP as the standard deviation of the data increases. Thus when comparing correlation results for analytical methods, one must consider carefully the standard deviation of the analyte values for the samples used in order to make a fair comparison. For the example shown, the SEE is set to 0.10, while the correlation is scaled from 0.0 to 1.0 for Sr values from 0.10 to 4.0. 

Van den Berg [131] used this technique to determine nanomolar levels of nitrate in seawater. Samples of seawater from the Menai Straits were filtered and nitrite present reacted with sulfanilamide and naphthyl-amine at pH 2.5. The pH was then adjusted to 8.4 with borate buffer, the solution de-aerated, and then subjected to absorptive cathodic stripping voltammetry. The concentration of dye was linearly related to the height of the reduction peak in the range 0.3-200 nM nitrate. The optimal concentrations of sulfanilamide and naphthyl-amine were 2 mM and 0.1 mM, respectively, at pH 2.5. The standard deviation of a determination of 4 nM nitrite was 2%. The detection was 0.3 nM for an adsorption time of 60 sec. The sensitivity of the method in seawater was the same as in fresh water. 

The limit of quantitation (LoQ) is the lowest concentration of analyte that can be determined with an acceptable level of uncertainty. This should be established by using an appropriate reference material or sample. It should not be determined by extrapolation. Various conventions take the approximate limit to be 5, 6 or 10 times the standard deviation of a number of measurements made on a blank or a low-level spiked solution. 

Let us suppose that the maximum allowable concentration of a contaminant in a workplace is 1 ppm. Twenty air samples are examined in order to establish if the mean contaminant concentration exceeds the allowable value. The mean and standard deviation of the sample are x = 1.9 ppm and s = 1.6 ppm. 

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. 

Kimura and Miller [29] demonstrated (Table 13.7) that mercury in several organic forms can be digested and aerated from unfiltered soil digests. For samples of lOg of soil cores containing 5pg mercury or less, the standard deviations of a single determination were 0.12, 0.15 and 0.23pg, respectively, using 2cm cylindrical optical cells. 

The precision, or reproducibility, of a method is the extent to which a number of replicate measurements of a sample agree with one another and is affected by the random error of the method. It is measured as imprecision, which is expressed numerically in terms of the standard deviation of a large number of replicate determinations (i.e. greater than 30), although for simplicity in the calculation shown in Procedure 1.1 only a limited number of replicates are used. The value quoted for s is a measure of the scatter of replicate measurements about their mean value and must always be quoted relative to that mean value. 

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

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 averages of random samples of a population are normally distributed. Therefore, the standard deviation of the population of sample means is the standard deviation of the population from which the sample is drawn divided by the square root of sample size. If we standardize the data to have a mean of 0.0 and a standard deviation of 1.0, then the standard deviation of the sample mean is 1.0 divided by the square root of the sample size. To be 95 percent confident that the incidence of insomnia in one group is smaller than the incidence in another group, the incidence in the first must be at least 1.64 standard deviations smaller than the incidence in the second. The sample size required to detect any given difference in means is approximately the square of 1.64 divided by the difference—in this case, (1.64/0.05) or 1,075.84. 

The mean weight and standard deviation of ten samples from each batch were determined for weight variation. Variation in diameter of beads was determined using a micrometer. The mean diameter (n=6) and the standard deviation were calculated for each batch. 

In addition, the standard deviation as calculated from a sample may sometimes be used as an estimate of the true standard deviation of a method or process. In these situations, tt is found that the standard deviation of a small sample tends to underestimate the true standard deviation. This bias can be compensated for by using one less than the number of observations as divisor of the sum of the squared deviations as given above. 

Solution (a) First compute the mean for the blanks and the standard deviation of the samples. Retain extra, insignificant digits to reduce round-off errors. 

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