Measurement of spread

Variance Another common measure of spread is the square of the standard deviation, or the variance. The standard deviation, rather than the variance, is usually reported because the units for standard deviation are the same as that for the mean value. 

Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. Eirst, does our measure of central tendency agree with the true, or expected value Second, why are our data scattered around the central value Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 

Evaluating Indeterminate Error Although it is impossible to eliminate indeterminate error, its effect can be minimized if the sources and relative magnitudes of the indeterminate error are known. Indeterminate errors may be estimated by an appropriate measure of spread. Typically, a standard deviation is used, although in some cases estimated values are used. The contribution from analytical instruments and equipment are easily measured or estimated. Indeterminate errors introduced by the analyst, such as inconsistencies in the treatment of individual samples, are more difficult to estimate. 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

Tlie expected value of a random variable X is also called "the mean of X" and is often designated by p. Tlie expected value of (X-p) is called die variance of X. The positive square root of the variance is called die standard deviation. Tlie terms and a (sigma squared and sigma) represent variance and standard deviadon, respectively. Variance is a measure of the spread or dispersion of die values of the random variable about its mean value. Tlie standard deviation is also a measure of spread or dispersion. The standard deviation is expressed in die same miits as X, wliile die variance is expressed in the square of these units. 

In tlie case of a random sample of observations on a continuous random variable assumed to have a so-called nonnal pdf, tlie graph of which is a bellshaped curve, tlie following statements give a more precise interpretation of the sample standard deviation S as a measure of spread or dispersion. 

The gaussian distribution is a good example of a case where the mean and standard derivation are good measures of the center of the distribution and its spread about the center . This is indicated by an inspection of Fig. 3-3, which shows that the mean gives the location of the central peak of the density, and the standard deviation is the distance from the mean where the density has fallen to e 112 = 0.607 its peak value. Another indication that the standard deviation is a good measure of spread in this case is that 68% of the probability under the density function is located within one standard deviation of the mean. A similar discussion can be given for the Poisson distribution. The details are left as an exercise. 

The standard deviation is very sensitive to outliers if the data are skewed, not only the mean will be biased, but also s will be even more biased because squared deviations are used. In the case of normal or approximately normal distributions,, v is the best measure of spread because it is the most precise estimator for standard deviation is often uncritically used instead of robust measures for the spread. 

The above measures of spread are expressed in the same unit as the data. If data with different units should be compared or the spread should be given in percent of the central value it is better to use a measure which is dimension free. Such a measure is... 

In the same way as the SD is used as a measure of spread around a mean, the SEM is used as a measure of the spread of a group of sample means around the true population mean. It is used to predict how closely the sample mean reflects the population mean. 

For a log-normal distribution, o-g is still a measure of spread of the distribution, but it has a slightly different definition because In D, rather than D, is assumed to have a normal distribution. For a log-normal distribution o-g is defined by Eq. (E) ... 

The mean summarises only one aspect of a distribution. We also need some measure of spread or dispersion, the tendency for observations to depart from the central tendency. The standard measure of dispersion is the variance ... 

Fig. 6.2. Decrease of mobility with increasing e, as observed in rubrene single crystal FET with different gate insulators. The bars give a measure of spread of mobility values. Inset ...

A second and orthogonally equivariant approach to robust PCA uses projection pursuit (PP) techniques. These methods maximize a robust measure of spread to obtain consecutive directions on which the data points are projected. In Hubert et al. [46], a projection pursuit (PP) algorithm is presented, based on the ideas of Li and Chen [47] and Croux and Ruiz-Gazen [48], The algorithm is called RAPCA, which stands for reflection algorithm for principal components analysis. 

The measure of spread, N, of the size distribution is also shown in Figure 7. At the exit of the atomizer, there is a high degree of poly-... 

The standard deviation and geometric standard deviation are statistical measures of spread. The former is more commonly used with powders having a narrow size range and is the difference between the 50% and the (50 I6)% sizes. The latter is more commonly used with powders having a wide size range and is the ratio of the 50% and the non-fractional (50 16)%sizes. The standard deviation is defined as ... 

Standard deviation (s.d.) The square root of the variance, a single measure of spread. 

Trophic level Position in the food chain, e.g., herbivore, carnivore, bottom-feeder. Ultrasonic cleaner Lab equipment using ultrasound in liquid bath for cleaning. Variance A single measure of spread or range in ratio data. 

The dynamic sorption apparatus clearly demonstrates the advantages of performing dynamic rather than static measurements of spreading and penetration at microscopic resolutions on substrates as complex as paper. The ability to observe ink or toner/paper interactions in situ should be especially valuable for investigating other non-impact technologies. 

Frequently, when a series of replicate analyses is performed, one of the results will appear to differ markedly from the others. A decision will have to be made whether to reject the result or to retain it. Unfortunately, there are no uniform criteria that can be used to decide if a suspect result can be ascribed to accidental error rather than chance variation. It is tempting to delete extreme values from a data set because they will alter the calculated statistics in an unfavorable way, that is, increase the standard deviation and variance (measures of spread)j and they may substantially alter the reported mean. The only reliable basis for rejection occurs when it can be decided that some specific error may have been made in obtaining the doubtful result. No result should be retained in cases where a known error has occurred in its collection. 

Variance. A measure of variability which is defined unambiguously on a population of values as the mean of the squared deviations of the observations from the mean. (The term is due to R.A. Fisher.) For a sample variance, however, two different conventions are observed. One is to take the sum of the squared deviations from the mean and divide by the number of observations the other is to divide the same sum by one less than the number of observations. The variance is a statistic of considerable theoretical importance. For a sample of observations from a Normal distribution, when taken together with the mean, it provides a complete summary of all relevant features of the data. As a statistic, however, it is rather vulnerable to bad values and various alternative measures of spread are sometimes used. 

The Bloomberg ASW screen shows the Z-spread. It is 46.1 for HERIM and 45.9 for TKAAV. The Z-spread provides hence a better measure of spread, although giving a similar result in terms of investor s decision. However, being a constant measure, it does not consider the timing of default In fact, each cash flow has a different level of credit risk. To overcome this hmitatirHi, the Z-spread spread could be adjusted by introducing a probability of default for each cash flow. This other spread is referred to adjusted Z-spread or C-spread. 

The positive square root of the variance is called the standard deviation, or AA. The standard deviation is the most commonly used measure of spread, and we shall take it as the measure of the uncertainty in the property A. 

There are several different measures of spread the two most common ones are spread to benchmark and asset swap spread/margin (ASM). The former is the difference in the yields of a bond and its corre-... 

Another measure of spread is called the quartile in spreadsheet. If the data are sorted from lowest to highest, quartiles can be established in a spreadsheet ... 

Another widely used measure of spread is the coefficient of variation (CV), also known as the relative standard deviation (RSD), which is given by 100s/3f. 

In experimental science we normally have no advance knowledge of whether or not our data might come from a heavy-tailed distribution or might contain outliers. So ideally we would like to use an estimate of location (the mean and the median are location estimates) that behaves like the mean when the underlying distribution is truly normal, but has the robust properties of the median when outliers or heavy tails do occur. Analogous arguments apply to measures of spread. Over 30 years ago Huber and others showed that these desirable properties are available. 

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