Frequency distribution curves

The different MW averages are derived by using the differential or frequency distribution curves and are reported as number average MW (Mn), weight average... 

The shape of a frequency distribution curve will depend on how the size increments were chosen. With the common methods for specifying increments, the curve will usually take the general form of a skewed probability curve with a single peak. However, it may also have multiple peaks, as in Fig 2, There are various analytical relationships for representing size distribution. One or the other may give a better fit of data in a particular instance. There are times, however, when analytical convenience may justify one. The log-probability relationship is particularly useful in this respect... 

Fig. 2 Normal, or Gaussian, size-frequency distribution curve. Percentage of particles lying within 1 and 2 standard deviations about the arithmetic mean diameter are indicated.

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

Figure 4.1. (a) Histograms and (b) frequency distribution curves of droplet sizes based on number and volume. 

The quantal dose-response curve is actually a cumulative plot of the normal frequency distribution curve. The frequency distribution curve, in this case relating the minimum protective dose to the frequency with which it occurs in the population, generally is bell shaped. If one graphs the cumulative frequency versus dose, one obtains the sigmoid-shaped curve of Figure 22A. The sigmoid shape is a characteristic of most dose-response curves when the dose is plotted on a geometric, or log, scale. 

FIGURE 9.13 Frequency distribution curve (logarithmic size scale) (adapted from Hinds, 1982). 

Fig. 26.1 Relative frequency distribution curves obtained during field evaluation of a competitive ELISA for drug residues the area represented by (A) contains the true-positive results (D) true-negative results (B) false-positive results (C) false-negative results.

This matter has been discussed by Polanyi and Wigner.f Plotting a frequency distribution curve for the values of the constant multiplying the exponential term, for reactions in solution as well as in the gaseous state, they obtain fig. 4. 

Figure 7.6 Frequency distribution curves showing (A) the normal variability in plasma concentration when a fixed dose of drug X is administered to a large population of patients and (B) a bimodal distribution typical of a pharmacogenetic alteration when drug Y is administered under the same conditions.

It is not possible to derive the frequency distribution curves from the curves reported in the Figure 11. The only value of these demonstrations is that they uncover experimental criteria which are useful for examining a model describing the methyl group motion. 

Data available from past e3q>erience can be used to generate frequency distribution curves. 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 curves. It is most unlikely that any frequency distribution curve obtained in practice will exactly fit a curve 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 curve can be used to plot a cumulative-frequency curve. This is the curve of most importance in business decisions and can be plotted from a normal frequency distribution curve (see Sec. 3). The cumulative curve represents the probability of a random value z having a value of, say, Zi 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 curve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probability of occurrence. The cumulative probability 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. 

The underlying frequency distribution curve of the Gompertz curve may be obtained by diflFerentiation of Eq. (9-96) to give... 

Since d l -y) = -dy, the corresponding underlying frequency distribution curve is from Eq. (9-100) ... 

Fig. 2 Frequency distribution curves corresponding to (A) a normal distribution (B) a positively skewed distribution and (C) a bimodal distribution.

Q10 (temperature coefficient) The increase in the rate of a chemical process due to raising the temperature by 10 C. quantal responses Are all-or-none responses, or qualitative responses, e.g. death or survival (in contrast to quantitative responses which are continuous variables). The underlying distribution is the binomial distribution. Log dose-response lines for quantal responses are frequently sigmoidal in shape, and since this is the same form as the integrated frequency distribution curve, the slope of the... 

Figure 14-15 Cumulative frequency distribution of relative differences for the comparison of drug assays example.The lighter curve indicates the Gaussian cumulative frequency distribution curve. In accordance with the test for normality, a good agreement is observed.

Consider the set of data below, which represents polychlorinate biphenyl (PCB) levels in a contaminated water stream for a given hour for 25 days. As a first step in summarizing the data, you are requested to form a frequency table, a frequency polygon, a cumulative frequency table, and a cumulative frequency distribution curve. 

Figure 130. Histogram of percent frequency distribution curve.

Fig 3.7 The probability is proportional to the surface under the frequency distribution curve. 

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