Sampling population standard deviation

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. 

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. 

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. 

The sample standard deviation, s, provides an estimate of the population standard deviation, a. The (n — 1) term in equations (6.4) and (6.6) is often described as the number of degrees of freedom (frequently represented in statistical tables by the parameter v (Greek letter, pronounced nu ). It is important for judging the reliability of estimates of statistics, such as the standard deviation. In general, the number of degrees of freedom is the number of data points (n) less the number of parameters already estimated from the data. In the case of the sample standard deviation, for example, v = n — 1 since the mean (which is used in the calculation of s) has already been estimated from the same data. 

The standard deviation is used to describe the dispersion of individual measurement results. If we make a number of repeated measurements on the same sample, the standard deviation provides an estimate of the expected spread of the results. The standard deviation of the mean describes the dispersion of mean values estimated from a number of samples drawn at random from the same population of data. The standard deviation of the mean will always be smaller than the standard deviation by a factor of +Jn, where n is the number of values that have been averaged to obtain the estimate of the mean. 

The term r-i, is not a parameter of the model but is a single value sampled from the population of possible deviations [Natrella (1963)]. The magnitude of r-i, might be used to provide an estimate of a parameter associated with that population of residuals, the population variance of residuals, aj. The population standard deviation of residuals is a,. The estimates of these two parameters are designated s] and s respectively [Neter, Wasserman, and Kutner (1990)]. If DF, is the number of degrees of freedom associated with the residuals, then... 

Normally the population standard deviation a is not known, and has to be estimated from a sample standard deviation s. This will add an additional uncertainty and therefore will enlarge the confidence interval. This is reflected by using the Student-t-distribution instead of the normal distribution. The t value in the formula can be found in tables for the required confidence limit and n-1 degrees of freedom. 

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 ... 

Population Standard Deviation The dispersion of data around the mean for the entire population of possible samples (an infinite number of samples), which is approximated by n > 20, is called the population standard deviation and is given the symbol o (Greek letter sigma). 

Population standard deviation—standard deviation of sample—s ... 

Transform the data first by taking logarithms and then standardising over die 14 training set samples (use the population standard deviation). Why are these transformations used ... 

Standard deviations, variances and covariances are useful common functions. It is important to recognise that there are both population and sample functions, so that STDEV is the sample standard deviation and STDEVP the equivalent population standard deviation. Note that for standardising matrices it is a normal convention to use the population standard deviation. Similar comments apply to VAR and VARP. 

The mean function can be used in various ways. By default this function produces die mean of each column in a matrix, so that mean (W) results in a 1 x 3 row vector containing die means. It is possible to specify which dimension one wishes to take die mean over, the default being die first one. The overall mean of an entire matrix is obtained using the mean function twice, i.e. mean (mean (W) ). Note that the mean of a vector is always a single number whether the vector is a column or row vector. This function is illustrated in Figure A.39. Similar syntax apphes to functions such as min, max and std, but note that the last function calculates the sample rather dian population standard deviation and if employed for scaling in chemometrics, you must convert back to the sample standard deviation, in the current case by typing std(W) /sqrt ( (s (1) ) / (s (1) -1) ), where sqrt is a function that calculates the square root and s contains the number of rows in die matrix. Similar remarks apply to the var function, but it is not necessary use a square root in the calculation. 

Random errors Relative standard deviation Robust variance Samples and populations Standard deviation of the mean (standard error of the mean)... 

Note that the term (n - 1) is used in the denominator of this equation to ensure that 5 is an unbiased estimate of the population standard deviation, a. (There is a general convention in statistics that English letters are used to describe the properties of samples, Greek letters to describe populations). The term (n - 1) is the number of degrees of freedom of the estimate, s. This is because if x is known, it is only necessary to know the values of (n — 1) of the individual measurements, as by definition S(.x,- — x) = 0. The square of the standard deviation, is known as the variance, and is a very important statistic when two or more sources of error are being considered, because of its additivity properties. 

In most instances the population standard deviation is not known and must be estimated from the sample standard deviation (s). Substitution of s for cr in equation (1.1) with M = 1.96 does not result in a 95% confidence interval unless the sample number is infinitely large (in practice >30). When s is used, multipliers, whose values depend on sample number, are chosen from the /-distribution and the denominator in... 

Most practical exercises are based on a limited number of individual data values (a sample) which are used to make inferences about the population from which they were drawn. For example, the lead content might be measured in blood samples from 100 adult females and used as an estimate of the adult female lead content, with the sample mean (T) and sample standard deviation (j) providing estimates of the true values of the underlying population mean (/r) and the population standard deviation (c). The reliability of the sample mean as an estimate of the true (population) mean can be assessed by calculating the standard error of the sample mean (often abbreviated to standard error or SE), from ... 

For a finite sample of n observations, the sample mean x is the arithmetic average of the n observations x approaches the population mean p in the limit as n approaches infinity. Correspondingly, the sample standard deviation s approaches the population standard deviation a in the limit. The sample standard deviation is given by the equation... 

A difficulty is that the population standard deviation is not usually known and can only be approximated for a finite number of measurements by the sample standard deviation s, calculated from (26-4). This difficulty is overcome for gaussian distributions by use of the quantity t (sometimes known as Student s t), defined by... 

In Chapters 11 and 12, the symbol o is used for the population standard deviation (i.e., when the sample size is large) and the symbol s for the sample standard deviation (when the sample size is small). 

The quantity Sp is the pooled estimate of the parent population standard deviation and, for equal numbers of samples in the two sets ( i = /I2), is given by... 

Figure 6-4a shows two Gaussian curves in which we plot the relative frequency y of various deviations from the mean versus the deviation from the mean. As shown in the margin, curves such as these can be described by an equation that contains just two parameters, the population mean p. and the population standard deviation a. The term parameter refers to quantities such as pu and a that define a population or distribution. This is in contrast to quantities such as the data values x that are variables. The term statistic refers to an estimate of a parameter that is made from a sample of data, as discussed below. The sample mean and the sample standard deviation are examples of statistics that estimate parameters p. and a, respectively. 

Equation 6-4 applies to small sets of data. It says, Find the deviations from the mean d square them, sum them, divide the sum by A/ - 1, and take the square root. The quantity A/ - 1 is called the number of degrees of freedom. Scientific cal culators usually have the standard deviation function built in. Many can find the population standard deviation o- as well as the sample standard deviation, s. For any small data set, you should use the sample standard deviation, s. 

When you make statistical calculations, remember that because of the uncertainty in X, a sample standard deviation may differ significantly from the population standard deviation. As N becomes larger, x and s become better estimators of and cr. 

If we have several subsets of data, we can get a better estimate of the population standard deviation by pooling (combining) the data than by using only one data set. Again, we must assume the same sources of random error in all the measurements. This assumption is usually valid if the samples have similar compositions and have been analyzed in exactly the same way. We must also assume that the samples are randomly drawn from the same population and thus have a common value of a. 

The International Union of Pure and Applied Chemistry recommends that the symbol r, be used for relative sample standard deviation and cr for relative population standard deviation. In equations where it is cumbersome to use RSD, we will use and cr. ... 

The size of the confidence interval, which is computed from the sample standard deviation, depends on how well the sample standard deviation s estimates the population standard deviation cr. If is a good approximation of a, the confidence interval can be significantly narrower than if the estimate of a is based on only a few measurement values. 

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