Population, mean and standard deviation

If X and a are good estimates of the population mean and standard deviation then z will be approximately normally distributed with a mean of zero and unit standard deviation. An analytical result is described as well behaved when it complies with this condition. 

Simulation. One approach is to assume the sample mean and standard deviation are the true population mean and standard deviation, to provide a best estimate of the true probability of passing. This has the advantage that it can provide estimates of the probability of passing at any stage and can handle the nonsymmetric potency shelf life limits in the content uniformity test. The disadvantage is that it does not provide a bound on the probability with high assurance and is not a function of sample size. It can provide a good summary statistic of the content uniformity data, however. 

The population mean and standard deviation cannot be determined for an infinite population hence, they must be estimated from a sample of size n. When fi and a are estimated from small samples, fi x and o s, the uncertainty in their estimates may be large, depending on the size of n, thus the confidence interval described in Equation 3.8 must be inflated accordingly by use of the /-distribution. When n is small, say 3 to 5, the uncertainty is large, whereas when n is large, say 30 to 50, the uncertainty is much smaller. 

In a situation with many data from each individual drawn in an inter subject balanced manner, a two stage method is very often used each individual is fitted individually without considering the inter individual dependencies. In a second step, the parameters are resumed as population mean and standard deviation, often considered as inter... 

These occur when random events act to produce variability in a continuous characteristic (quantitative variable). This situation occurs frequently in chemistry, so normal distributions are very useful and much used. The bell-hke shape of normal distributions is specified by the population mean and standard deviation (Fig. 41.4) it is symmetrical and configured such that 68.27% of the data will lie within 1 standard deviation of the mean. 95.45% within 2 standard deviations of the mean, and 99.73% within 3 standard deviations of the mean. 

Figure 8-15 is a typical instalment control chart for an analytical balance. Mass data were collected on 20 consecutive days for a 20.000-g standard mass certified by the National Instimte of Standards and Technology. On each day, five replicate determinations were made. From independent experiments, estimates of the population mean and standard deviation were found to be = 20.000 g and... 

The unknown quantities of interest described in the previous section are examples of parameters. A parameter is a numerical property of a population. One may be interested in measures of central tendency or dispersion in populations. Two parameters of interest for our purposes are the mean and standard deviation. The population mean and standard deviation are represented by p and cr, respectively. The population mean, p, could represent the average treatment effect in the population of individuals with a particular condition. The standard deviation, cr, could represent the typical variability of treatment responses about the population mean. The corresponding properties of a sample, the sample mean and the sample standard deviation, are typically represented by x and s, which were introduced in Chapter 5. Recall that the term "parameter" was encountered in Section 6.5 when describing the two quantities that define the normal distribution. In statistical applications, the values of the parameters of the normal distribution cannot be known, but are estimated by sample statistics. In this sense, the use of the word "parameter" is consistent between the earlier context and the present one. We have adhered to convention by using the term "parameter" in these two slightly different contexts. 

When you have a lot of data Confidence interval knowing the population mean and standard deviation... 

The average and sample standard deviation are known as estimators of the population mean and standard deviation. We have seen how the estimates improve as the number of data increases. As we have stressed, the use of these statistics requires data that are normally distributed, and for confidence intervals employing the standard deviation of the mean this tends to be so. Real data may be so distributed, but often the distribution will contain data that are seriously flawed, as with the RACI titration competition described in chapter 1. If we can identify such data and remove them from further... 

Statisticians normally use Latin sjrmbols to represent sample values, reserving the Greek alphabet for population parameters. Following this convention, we represent the population mean and standard deviation of our example by the Greek letters p and a, respectively. What can we infer about the values of these parameters, knowing the sample values xand s ... 

The designer knows very well that the equations used for deriving / (99.9) and R 99.99) probability of survival are valid for a total population mean and standard deviation while the values he calculated on the sample of 15 specimens depend on the sample size. To check that he runs some 15 new traction tests on cylindrical specimens of the same size. The results are listed in the previous table. Using the new sample of 30 specimens he re-run aU the statistics and gets... 

In this problem you will collect and analyze data in a simulation of the sampling process. Obtain a pack of M M s or other similar candy. Obtain a sample of five candies, and count the number that are red. Report the result of your analysis as % red. Return the candies to the bag, mix thoroughly, and repeat the analysis for a total of 20 determinations. Calculate the mean and standard deviation for your data. Remove all candies, and determine the true % red for the population. Sampling in this exercise should follow binomial statistics. Calculate the expected mean value and expected standard deviation, and compare to your experimental results. 

In the above ealeulations of the mean, varianee and standard deviation, we make no prior assumption about the shape of the population distribution. Many of the data distributions eneountered in engineering have a bell-shaped form similar to that showed in Figure 1. In sueh eases, the Normal or Gaussian eontinuous distribution ean be used to model the data using the mean and standard deviation properties. 

One drag level (Cindiv) can be used with the means and standard deviation (SD) of population parameters (Ppop) as a priori knowledge for an individual parameter estimate using the Bayesian objective function. 

A basic requirement, in order that the above results mean and standard deviation are truly representative of the sampled population, is that the individual measurements should be independent of each other. Two general cases must be distinguished ... 

Note. If the N dimensions yield very different numerical values, such as 105 3 mmol/L, 0.0034 0.02 meter, and 13200 600 pg/ml, the Euclidian distances are dominated by the contributions due to those dimensions for which the differences A-B, AS, or BS are numerically large. In such cases it is recommended that the individual results are first normalized, i.e., x = (x - Xn,ean)/ 5 t, where Xmean and Sx are the mean and standard deviation over all objects for that particular dimension X, by using option (Transform)/(Normalize) in program DATA. Use option (Transpose) to exchange columns and rows beforehand and afterwards The case presented in sample file SIEVEl.dat is different the individual results are wt-% material in a given size class, so that the physical dimension is the same for all rows. Since the question asked is are there differences in size distribution , normalization as suggested above would distort tbe information and statistics-of-small-numbers artifacts in the poorly populated size classes would become overemphasized. 

Percentage means and standard deviations, except for intermediate populations where percentage values only are available. 

Greek letter, pronounced mu ), while its spread is characterized by the parameter a (Greek letter, pronounced sigma ), as shown in Figure 6.4. The parameters p and a are known as the mean and standard deviation of the population, respectively. 

As mentioned in Section 6.1.1, analysts generally have only a sample of data from a much larger population of data. The sample is used to estimate the properties, such as the mean and standard deviation, of the underlying population. 

The number of specimens required per composite was the minimum value of N satisfying Equation 1 for all compounds addressed by the broad scan analyses. One constraint on the value of N was that the total mass of the composited sample could not exceed sample preparation and extraction constraints (approximately 30 gram). To actually determine the value of N it was noted that the value of N satisfying Equation 1 depends on the assumed distribution of population residue levels. Various lognormal distributions (i.e. values of the mean and standard deviation) were therefore investigated to determine how N varies with these parameters. 

The calculation of mean and standard deviation only really makes sense when we are dealing with continuous, score or count data. These quantities have little relevance when we are looking at binary or ordinal data. In these situations we would tend to use proportions in the various categories as our summary statistics and population parameters of interest. 

Figure 4-5 illustrates the meaning of confidence intervals. A computer chose numbers at random from a Gaussian population with a population mean (p.) of 10 000 and a population standard deviation (o) of 1 000 in Equation 4-3. In trial 1, four numbers were chosen, and their mean and standard deviation were calculated with Equations 4-1 and 4-2. The 50% confidence interval was then calculated with Equation 4-6, using t = 0.765 from Table 4-2 (50% confidence, 3 degrees of freedom). This trial is plotted as the first point at the left in Figure 4-5a the square is centered at the mean value of 9 526, and the error bar extends from the lower limit to the upper limit of the 50% confidence interval ( 290). The experiment was repeated 100 times to produce the points in Figure 4-5a. 

Computer programs accounted for the presence of oil drops below- the detection limit of the Coulter Counter. The data processing procedure, which assumed that the oil-drop size distribution was lognormal, yielded accurate estimates of the true mean and standard deviation describing the emulsion drop size distribution. The data-analysis procedure did not affect the actual measured drop populations which were used in the kinetic studies. The computer programs are described in detail by Bycscda.8... 

The processes may be studied quantitatively by comparing the means and standard deviations of the two populations. The effect of final blend time on lubricant distribution was examined by comparing disintegration time statistics for the grouped data. None was noted. 

Probabilistic sampling, which lies in the core of the DQO process, is based on a random sample location selection strategy. It produces data that can be statistically evaluated in terms of the population mean, variance, standard deviation, standard error, confidence interval, and other statistical parameters. The EPA provides detailed guidance on the DQO process application for the... 

The ISO 5725 standard was used to interpret the data. Even if the main purpose of this standard is related to the validation of a method, it can be used to evaluate some components of the measurement uncertainty. The homogeneity of the population of results, in terms of mean and standard deviation was determined using statistical tests (Cochran and Grubbs). A few laboratories were rejected after the tests. Tables 3 and 4 present the comparison of overall performance of laboratories when working with usual and metrological calibrations solutions. 

Very often it is not possible a priori to separate contaminated and uncontaminated soils at the time of sampling. The best that can be done in this situation is to assume the data comprise several overlapping log-normal populations. A plot of percent cumulative frequency versus concentration (either arithmetic or log-transformed values) on probability paper produces a straight line for a normal or log-normal population. Overlapping populations plot as intersecting lines. These are called broken line plots and Tennant and White (1959) and Sinclair (1974) have explained how these composite curves may be partitioned so as to separate out the background population and then estimate its mean and standard deviation. Davies (1983) applied the technique to soils in England and Wales and thereby estimated the upper limits for lead content in uncontaminated soils. 

This rather complicated equation can be interpreted as follows. The function f (x) is proportional to the probability that a measurement has a value v for a normally distributed population of mean /< and standard deviation a. The function is scaled so that the area under the normal distribution curve is 1. 

A second facility that is sometimes useful is the random number generator function. There are several possible distributions, but the most usual is the normal distribution. It is necessary to specify a mean and standard deviation. If one wants to be able to return to the distribution later, also specify a seed, which must be an integer number. Figure A. 15 illustrates the generation of 10 random numbers coming from a distribution of mean 0 and standard deviation 2.5 placed in cells A1 -A10 (note that the standard deviation is of the parent population and will not be exactly the same for a sample). This facility is very helpful in simulations and can be employed to study die effect of noise on a dataset. 

Often it is of interest in dealing with a normal population to determine what percentage of the observations can be expected to fall within some specified interval when the mean and standard deviation are not known. 

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