Standard deviation of a population

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. 

Standard deviation — (of a population of data) (a) A measure of the -> precision of a population of data. It is the positive square root of the sum of the squares of the deviations between the observations and the - population mean (p), divided by the total number of replicate... 

The Gaussian distribution has a symmetrical, bell shape and is sufficiently characterized by its mean (/jl) and variance (cr2). Its standard deviation (cr) is the square root of the variance. The symbols pi and cr refer to the mean and standard deviation of a population (i.e., the set of all possible measurements of a particular quantity). In practice, /x, and cr often are not known because a population is too large to sample in its entirety. A subset of measurements of a particular quantity represents a sample of a population. If the sample is drawn from a normal population, the parameters characterizing the sample distribution are the sample mean of the measurements or observations (y) and the sample standard deviation (s). Both of these parameters are calculated based on the values of the sample observations and the number of observations,... 

Because of area relationships such as these, the standard deviation of a population of data is a useful predictive tool. For example, we can say that the chances are 68.3 in 100 that the random uncertainty of any single measurement is no more than 1(T. Similarly, the chances are 95.4 in 100 that the error is less than 2cr, and so forth. The calculation of areas under the Gaussian curve is described in Feature 6-2. 

The Greek letter used to indicate the standard deviation of a population, defined as the square root of the variance, e.g., for the normal (Gaussian) distribution ... 

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

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. 

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. 

The 206Pb/204Pb ratios of four samples from a Polynesian island have been determined to be 18.999, 19.091, 19.216, and 19.222. Assuming that these measurements represent a sample from a normal population, find a 95 percent confidence interval for the mean and the standard deviation of the population. 

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

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 quantity a is called the standard deviation of the population and is a measure of the spread or dispersion of the values about the mean, x (see Table 6.3). The smaller the value of a, the narrower the spread, a is calculated from... 

Standard Deviation Prediction Interval (SDPI). Since uniformity is of primary interest in powder blend validation and because of a concern that a constant sampling error can occur, one approach is to base the criteria only on variability. The SDPI allows one to predict, from a sample of size n and with a specified level of assurance, an upper bound on the standard deviation of a future sample of size m from the same population. This approach is recommended in the PDA paper on blend uniformity [1],... 

Estimation of the limits of accuracy (deviation from a true or theoretical value) is not ordinarily attempted in coal analysis. Precision, on the other hand, is determined by means of cooperative test programs. Both repeatability, the precision with which a test can be repeated in the same laboratory, usually but not always by the same analyst using the same equipment and following the prescribed method(s), and reproducibility, the precision expected of results from different laboratories, are determined. Values quoted in test methods are the differences between two results that should be exceeded in only 5 out of 100 pairs of results, equal to 2-Jl times the standard deviation of a large population of results. 

The standard deviation of a set of data, usually given the symbol 5, is the square root of the variance. The difference between standard deviation and variance is that the standard deviation has the same units as the data, whereas the variance is in units squared. For example, if the measured unit for a collection of data is in meters (m) then the units for the standard deviation is m and the unit for the variance is m2. For large values of n, the population standard deviation is calculated using the formula ... 

Standard error of the mean o/n0 5, where o is the standard deviation of the population from which a random sample of size n was taken (usually o is estimated by s). 

We would then look np in the table of t for 9 degrees of freedom its value for the 5% level (95% chance of being right is equivalent to a 5% chance of being wrong). Here t is 2.26. Then the limits dzL on either side of the sample mean, where a = the standard deviation of the population and n is the number of individuals in the sample, are... 

The standard deviation of the sample is experimentally important as an estimate of the desired standard deviation of the population, which cannot actually be determined with a finite number of measurements. If a random sample is taken, the quantity s becomes a closer approximation to a as the size of the sample is increased. Likewise, the sample mean x becomes a closer estimate of the population mean p as the size of a random sample is increased. Although in many practical papers the standard deviation of a finite number of observations is represented by a, strictly speaking the symbol a should be reserved for the universe, or infinite population. For some purposes it is convenient to use the variance, which for the sample is or V. From the relation... 

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

As a group, the volunteers were above average In physical and mental qualifications, with a mean IQ near 110, good behavior records, and "normal" MMIPs with profiles generally within two standard deviations of the population mean on all scales. 

The true (unknown, hypothetical) value of a property of the set is often called a parameter and given a lower-case Greek letter symbol. For example, the standard deviation of the population is symbolized by the Greek sigma The corresponding property determined in die... 

At times, there is a need to compare the variances (or standard deviations) of two populations. For example, the normal t test requires that the standard deviations of the data sets being compared are equal. A simple statistical test, called the F test, can be used to test this assumption under the provision that the populations follow the normal (Gaussian) distribution. The F test is also used in comparing more than two means (see Section 7C) and in linear regression analysis (see Section 8C-2). 

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