Population standard deviation

Standard deviation (population standard deviation), a The square root of the variance, the population standard deviation represents the dispersion of the population. In the normal distribution, 68% of the distribution lies at the mean /u 1 a. (Section 1.8.2)... 

The mean (m) and the standard deviation (a) of the population of localized AE sources over each cell of the (ATi,AT2) histogram are calculated. 

We have already found that the probability function governing observation of a single event x from among a continuous random distribution of possible events x having a population mean p and a population standard deviation a is... 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

The standard deviation is the square root of the average squared differences between the individual observations and the population mean ... 

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

Alternatively, a confidence interval can be expressed in terms of the population s standard deviation and the value of a single member drawn from the population. Thus, equation 4.9 can be rewritten as a confidence interval for the population mean... 

The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. A single tablet is randomly selected, analyzed, and found to contain 245 mg of aspirin. What is the 95% confidence interval for the population mean ... 

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... 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

When the target population is segregated, or stratified, equation 7.5 provides a poor estimate of the amount of sample needed to achieve a desired relative standard deviation for sampling. A more appropriate relationship, which can be applied to both segregated and nonsegregated samples, has been proposed. ... 

This short paper describes a demonstration suitable for use in the classroom. Two populations of corks are sampled to determine the concentration of labeled corks. The exercise demonstrates how increasing the number of particles sampled improves the standard deviation due to sampling. 

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. 

The quantity of sample required comprises two parts the volume and the statistical sample size. The sample volume is selected to permit completion of all required analytical procedures. The sample size is the necessary number of samples taken from a stream to characterize the lot. Sound statistical practices are not always feasible either physically or economically in industry because of cost or accessibiUty. In most sampling procedures, samples are taken at different levels and locations to form a composite sample. If some prior estimate of the population mean, and population standard deviation. O, are known or may be estimated, then the difference between that mean and the mean, x, in a sample of n items is given by the following ... 

Assume that the table represents typical production-hne performance. The numbers themselves have been generated on a computer and represent random obseiwations from a population with I = 3.5 and a population standard deviation <7 = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supeiwisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. 

This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 90 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor. 

O = standard deviation of the population from which the sample has been drawn... 

The population from which the obsei vations were obtained is normally distributed with an unknown mean [L and standard deviation... 

The standard deviation. s° for the sample corresponds to the true standard deviation O for the whole population in the same way that the mean x of the sample corresponds to the arithmetic average [L for the whole population. Equation (9-70) can be written more compactly as... 

The random nature of most physieal properties, sueh as dimensions, strength and loads, is well known to statistieians. Engineers too are familiar with the typieal appearanee of sets of tensile strength data in whieh most of the individuals eongregate around mid-range and fewer further out to either side. Statistieians use the mean to identify the loeation of a set of data on the seale of measurement and the variance (or standard deviation) to measure the dispersion about the mean. In a variable x , the symbols used to represent the mean are /i and i for a population and sample respeetively. The symbol for varianee is V. The symbols for standard deviation are cr and. V respeetively, although a is often used for both. In this book we will always use the notation /i for mean and cr for the standard deviation. 

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. 

The sample represents a population (source) which, if normally distributed, has a mean of 1.767% and a standard deviation of 0.103%. This can be illustrated as shown in Fig. 32-2. 

The measure of variation in this population is given by the standard deviation of the population (a) ... 

So how does one infer that two samples come from different populations when only small samples are available The key is the discovery of the t-distribution by Gosset in 1908 (publishing under the pseudonym of Student) and development of the concept by Fisher in 1926. This revolutionary concept enables the estimation of ct ( standard deviation of the population) from values of standard errors of the mean and thus to estimate... 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

It is important to know that the Greek letters a and p refer to the standard deviation and mean respectively of a total population, whilst the Roman letters s and x are used for samples of populations, irrespective of the values of the population mean and the population standard deviation. 

A brief digression. In the language of statistics, the results for each of the stepped distributions in Figure 10-1 constitute a sample1 of the population that is distributed according to the continuous curve for the universe. A sample thus contains a limited number of x s taken from the universe that contains all possible z s. All simple frequency distributions are characterized by a mean and a variance. (The square root of the variance is the standard deviation.) For the population, the mean is u and the variance is a2. For any sample, the mean is x and the (estimate of) variance is s2. Now, x and s2 for any sample can never be as reliable as p and a2 because no sample can contain the entire population ir and s2 are therefore only the experimental estimates of g and cr2. In all that follows, we shall be concerned only with these estimates for simplicity s sake, we shall call s2 the variance. We have already met s—for example, at the foot of Table 7-4. 

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. 

The standard deviation, Sj, is the most commonly used measure of dispersion. Theoretically, the parent population from which the n observations are drawn must meet the criteria set down for the normal distribution (see Section 1.2.1) in practice, the requirements are not as stringent, because the standard deviation is a relatively robust statistic. The almost universal implementation of the standard deviation algorithm in calculators and program packages certainly increases the danger of its misapplication, but this is counterbalanced by the observation that the consistent use of a somewhat inappropriate statistic can also lead to the right conclusions. 

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