Sample variances

It is an estimation of the unknown true value p of an infinite population. We can also define the sample variance s as follows ... 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

The fact that each sample variance is related to its own population variance means that the sample variance being used for the calculation need not come from the same population. This is a significant departure from the assumptions inherent in the z, r, and statistics. 

To determine which step has the greatest effect on the overall variance, both si, and si must be known. The analysis of replicate samples can be used to estimate the overall variance. The variance due to the method is determined by analyzing a standard sample, for which we may assume a negligible sampling variance. The variance due to sampling is then determined by difference. 

The following data were collected as part of a study to determine the effect of sampling variance on the analysis of drug animal-feed formulations.2... 

The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 

Solving for n allows us to calculate the number of particles that must be sampled to obtain a desired sampling variance. 

Note that the relative sampling variance is inversely proportional to the number of particles sampled. Increasing the number of particles in a sample, therefore, improves the sampling variance. 

Suppose you are to analyze a solid where the particles containing analyte represent only 1 X 10 % of the population. How many particles must be collected to give a relative sampling variance of 1% ... 

Thus, to obtain the desired sampling variance we need to collect 1 X 10 particles. 

Treating a population as though it contains only two types of particles is a useful exercise because it shows us that the relative sampling variance can be improved by collecting more particles of sample. Furthermore, we learned that the mass of sample needed can be reduced by decreasing particle size without affecting the relative sampling variance. Both are important conclusions. 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

This is not an uncommon problem. For a target population with a relative sampling variance of 50 and a desired relative sampling error of 5%, equation 7.7 predicts that ten samples are sufficient. In a simulation in which 1000 samples of size 10 were collected, however, only 57% of the samples resulted in sampling errors of less than 5% By increasing the number of samples to 17 it was possible to ensure that the desired sampling error was achieved 95% of the time. 

Unfortunately, the simple situations just described are often the exception. In many cases, both the sampling variance and method variance are significant, and both multiple samples and replicate analyses of each sample are required. The overall error in this circumstance is given by... 

A certain analytical method has a relative sampling variance of 0.40% and a relative method variance of 0.070%. Evaluate the relative error (a = 0.05) if (a) you collect five samples, analyzing each twice and, (b) you collect two samples, analyzing each five times. 

As expected, since the relative method variance is better than the relative sampling variance, a sampling strategy that favors the collection of more samples and few replicate analyses gives the better relative error. 

A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeterminate errors, the variance, s, of the data in each column provides an estimate of O rand- A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s ), is calculated by summing the squares of the differences between the replicates for each sample and that sample s mean, and dividing by the degrees of freedom. 

Finally, the data for each analyst can be reduced to separate mean values, A . The variance of the individual means about the global mean is called the between-sample variance, s, and is calculated as... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

Example For two sample variances with 4 df each, what limits will bracket their ratio with a midarea prohahility of 90 percent ... 

Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact Cj = cf. There is not sufficient evidence to reject the null hypothesis that Of = <31. 

Since each treatment has tit experiments, the number of degrees of freedom is tit — 1. Then the sample variances are... 

Example 4 Calculation of Sample Weight for Surface Moisture Content An example is given with reference to material with minimal internal or pore-retained moisture such as mineral concentrates wherein physically adhering moisture is the sole consideration. With this simphfication, a moisture coefficient K is employed as miiltipher of nominal top-size particle size d taken to the third power to account for surface area. Adapting fundamental sampling theory to moisture sampling, variance is of a minimum sample quantity is expressed as... 

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