Cumulative Frequency plot

Figure 4.5 Cumulative frequency plot and determination of mean and standard deviation graphically...

Method III is called a "Histogram Plot" while Methods I n are called "Frequency Plot" and "Cumulative Frequency Plot", respectively. There is one important point which needs to be emphasized. That is ... 

A typical testing procedure involves several steps. First, the selected number and size of sieves are stacked upon one another, with the largest openings (inversely related to mesh per inch) being at the top of the stack, and beneath that a pan to collect the particles finer than the smallest sieve. The known amount of powder to be analyzed is then placed on the top sieve and the set is vibrated in a mechanical device for a predetermined time period. The results are obtained by weighing the amount of material retained on each sieve and on the collecting pan. The suction method uses one sieve at a time and examines the amount retained on the screen. In both methods the data are expressed as frequency or cumulative frequency plots, respectively. 

Figure 21.1 Cumulative frequency plot illustration 25th, 75th percentiles, and the interquartile range. estimate of the population variance. Population means and variances are by convention denoted by the Greek letters p and o2, respectively, while the corresponding sample parameters are denoted by X and s2.

Fig. 7—Cumulative frequency plot of nifedipine-Eudragit microspheres as a function of methylene chloride solution viscosity.

Fig. 9—Cumulative frequency plot of nifedipine-Eudragit microspheres as a function of the initial nifedipine concentration. Microspheres were prepared using 1 1 of 0.8% PVA and 25 g of Eudragit RS RL mixture (1 1) in 80 ml of methylene chloride stirred at 400 rev/min.

The traditional method of identifying a magnitude threshold has been accomplished by a variety of techniques. These include (1) the mean plus two standard deviations of a normally-distributed data set (2) arbitrarily selecting the 90 percentile or 95 percentile, etc., of the data (3) identifying the inflection point on a cumulative frequency plot that deviates from a straight line (Sinclair, 1976). 

Figures 4 and 5 show how ag may be derived from the cumulative frequency plots of the distribution of log x.

The residuals, if they are a measurement of the error, would be expected to be normally distributed. Therefore they may be expressed as a cumulative frequency plot, just as was done for the estimated coefficients in chapter 3. If the plot is a straight line, this supports the adequacy of the model. If there are important deviations this may indicate an inappropriate model, the need for a transformation, or errors in the data. The deviating points should be looked at individually. 

Methods I, II and III are called "Frequency Plot", "Cumulative Frequency Plot", and "Histogram respectively. 

A histogram or cumulative frequency plot will show what proportion of measurements exceed the criteria to show the extent of high corrosion risk. Where the ASTM criteria do not apply they will show the distribution of readings so that high risk areas can be identified. 

Figure 13.5 shows the same data as a cumulative frequency plot. The solid black line is the best fit normal distribution. If improved control halves the standard deviation (coloured line) then the mean can be increased by half the current average deviation from target, i.e. 

Figures 12-10 and 12-12 are examples of cumulative frequency plots for distributions that are normally distributed in volume. Values of dio, dso, and dgo (defined above) are readily determined for lOOFv equal to 10, 50, and 90, respectively. The slopes of the curves are a measure of the breadth of the distribution. A steeper slope means a narrower size distribution.

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

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. 

A common form of societal risk measure is an F-N curve, which is normally presented as a cumulative distribution plot of frequency F... 

Figures 12 and 13 illustrate two of the more commonly used methods for displaying societal risk results (1) an F-N curve and (2) a risk profile. The F-N curve plots the cumulative frequencies of events causing N or more impacts, with the number of impacts (N) shown on the horizontal axis. With the F-N curve you can easily see the expected frequency of accidents that could harm greater than a specified number of people. F-N curve plots are almost always presented on logarithmic scales because of...

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,... 

In Fig. 1 the cumulative frequency of the measured mean values for induvidual houses is plotted on a log-normal scale. The aritmetric mean value in our measurements is 160 Bq/nP. Areas with high concentrations are overrepresented in this distribution, (as seen from the figure) and by population weighing the distribution for the municipalities, a population weighted average of 110 Bq/m in the heating season is obtained. 

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. 

The results from the first study suggest rather clearly that conversation group sizes are limited at about four individuals (one speaker and three listeners) (Dunbar et al. 1995). Fig. 3 plots the cumulative frequency distributions for the number of individuals that a speaker can reach (i.e. conversation group size less one, since there is always only one speaker at any given moment per conversation Dunbar et al. 1995). All three datasets in the sample suggest that the number of listeners rapidly approaches an asymptotic value at around three. 

The characteristic curve of the odor threshold is used. The relative cumulative frequency of positive answers is calculated for each odorant concentration and graphically plotted, while for odor concentration a logarithmic scale is used. The odor threshold can be obtained from the resulting curve as the 50-percentile and so can the associated 16- and 84-percentiles. 

F-N curve—A plot of cumulative frequency verses consequences (often expressed as number of fatalities). A societal risk measure. 

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. 

Quantal dose-effect plots. Shaded boxes (and the accompanying bell-shaped curves) indicate the frequency distribution of doses of drug required to produce a specified effect that is, the percentage of animals that required a particular dose to exhibit the effect. The open boxes (and the corresponding colored curves) indicate the cumulative frequency distribution of responses, which are lognormally distributed. 

Plot of the cumulative frequency of craters versus crater diameter, for three geologic units on the Moon. Crater density measurements provide a means of ordering units in relative time. Modified from Neukum etal. (2001). 

Figure 1 shows, as a typical plot, the particle size distributions of the size fractions from a Johnie Boy sample. Only the first fraction, containing the largest particles, deviates significantly from lognormality. The standard deviations are almost the same for the first nine fractions as is apparent from the parallelism of the cumulative frequency curves. When the particle size decreases further, the standard deviations of the size distributions in the fractions increase. 

Very often it is not possible a priori to separate contaminated and uncontaminated soils at the time of sampling. The best that can be done in this situation is to assume the data comprise several overlapping log-normal populations. A plot of percent cumulative frequency versus concentration (either arithmetic or log-transformed values) on probability paper produces a straight line for a normal or log-normal population. Overlapping populations plot as intersecting lines. These are called broken line plots and Tennant and White (1959) and Sinclair (1974) have explained how these composite curves may be partitioned so as to separate out the background population and then estimate its mean and standard deviation. Davies (1983) applied the technique to soils in England and Wales and thereby estimated the upper limits for lead content in uncontaminated soils. 

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