Frequency distributions

The decision on how big the subgroup should be will depend on cost and obviously, destructive testing of an expensive product would be a limitation. 

A simple rule is to take the square root of the number of samples and round to the nearest whole number k = which in the above example is VTOO = 10. Alternatively, the sample size can be determined by reference to a table (Table 18.3). 

Rounding this number to the nearest value with the same number of decimal places as the original sample gives 77=0.06. 

Take the smallest measurement, 3.20 (round down if required), which is the low end of the first class limit now add the class width 3.20 + 0.06 = 3.26. 

Group Group range Number of tallies Group frequency  

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Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

The values of x and s vary from sample set to sample set. However, as N increases, they may be expected to become more and more stable. Their limiting values, for very large N, are numbers characteristic of the frequency distribution, and are referred to as the population mean and the population variance, respectively. 

The explanation of the hydrogen atom spectmm and the photoelectric effect, together with other anomalous observations such as the behaviour of the molar heat capacity Q of a solid at temperatures close to 0 K and the frequency distribution of black body radiation, originated with Planck. In 1900 he proposed that the microscopic oscillators, of which a black body is made up, have an oscillation frequency v related to the energy E of the emitted radiation by... 

The check sheet shown below, which is tool number five, is a simple technique for recording data (47). A check sheet can present the data as a histogram when results are tabulated as a frequency distribution, or a mn chart when the data are plotted vs time. The advantage of this approach to data collection is the abiUty to rapidly accumulate and analy2e data for trends. A check sheet for causes of off-standard polymer production might be as follows ... 

Fig. 3. Cumulative frequency distribution plotted by A, number B, surface area and C, volume, for the data in Table 1.

In reference to the tensile-strength table, consider the summary statistics X and. s by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution oft values summarized in Table 3-5. 

Data available from past experience can be used to generate frequency distribution cuiwes. It is essential for a company to have an efficient commercial-intelhgence system to assess market conditions. 

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

A frequency distribution cui ve can be used to plot a cumulative-frequency cui ve. This is the cui ve of most importance in business decisions and can be plotted from a normal frequency distribution cui ve (see Sec. 3). The cumulative cui ve represents the probability of a random value z having a value of, say, Z or less. 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a veiy flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped cui ves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the cui ve in the y axis to exist. The resulting single maximum cui ve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. 

When the data do not warrant the accuracy of Eq. (9-89) or Eq. (9-90), simpler cui ves will usually suffice if the frequency distribution may be assumed to have a single maximum value. 

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the cui ve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical beU-shaped cui ve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

A measure of risk to a group of people. It is most often expressed in terms of the frequency distribution of multiple casualty events... 

Figure 4.10 Cumulative frequency distribution for SAE 1018 yield strength data...

Figure 4.48 Load frequency distribution for the foot pedal...

We can compare the caicuiated 2-parameter Weibuii distribution with the originai frequency distribution by muitipiying the f x) or PDF by the scaiing factor, Nw, as determined from equation 4.86. The variate, x, is the mid-ciass vaiue over the ioad range. Aiso note that the popuiation, N, is divided by iO to change the frequency back to a percentage vaiue. The resuits of this exercise are shown in Figure 4.50. 

Figure 1 Histogram showing discontinuous or discrete frequency distribution...

Once the mean and standard deviation have been determined, the frequency distribution determined from the PDF can be compared to the original histogram, if one was constructed, by using a scaling factor in the PDF equation. For example, the expected frequency for the Normal distribution is given by ... 

Larsen (18-21) has developed averaging time models for use in analysis and interpretation of air quality data. For urban areas where concentrations for a given averaging time tend to be lognormally distributed, that is, where a plot of the log of concentration versus the cumulative frequency of occurrence on a normal frequency distribution scale is nearly linear,... 

The use of various statistical techniques has been discussed (46) for two situations. For standard air quality networks with an extensive period of record, analysis of residuals, visual inspection of scatter diagrams, and comparison of cumulative frequency distributions are quite useful techniques for assessing model performance. For tracer studies the spatial coverage is better, so that identification of meiximum measured concentrations during each test is more feasible. However, temporal coverage is more limited with a specific number of tests not continuous in time. 

Frequency distribution tables for wind speed and stability classes. 

I). iOtoO.459). Thejoint frequency distribution of responses at different critical locations in the piping is calculated. The convolution of the frequency distribution with the fragilities yields the conditional frequency of the initiating event. In preparing the list of the initiating events, it is necessary to consider the possibility lif multiple initiating events rather than single,... 

Each body having a temperate above absolute zero radiates energy in the form of electromagnetic waves. The amount of energy emitted is dependent on the temperature and on the emissivity of the material. The wavelength or frequency distribution (the spectrum) of the emitted radiation is dependent on the absolute temperature of the body and on the surface properties. 

Cumulati ve frequencies Accumulated sums of frequency values in a frequency distribution. 

Fig. 3 Frequency distribution of TLC/HPTLC publications over the years 1967 — 1986 (search made from Chemical Abstracts).

Manual transfer of the chromatographically separated substance to the detector . These include, for example, the detection of antibiotically active substances, plant and animal hormones, mycotoxins, insecticides, spice and bitter principles and alkaloids. The frequency distribution of their employment is shown in Figure 54 [295]. 

Fig. 54 Fields of application and frequency distribution of biological-physiological detection methods.

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