Cumulative frequency curve

When a cumulative-frequency curve can be satisfactorily represented by a logistics cuiwe, the underlying frequency cui ve can be obtained by differentiation of Eq. (9-91) as... 

Fig. 6.1 Activity histogram (left Y axis) and cumulative frequency curve (right Y axis) versus binned percent inhibition (X axis) for a recent primary HTS run at Wyeth. Note the sudden spike in the cumulative frequency curve for compounds with >100% inhibition, corresponding to the 0.2% most active compounds.

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. 

The geometric standard deviation (GSD) is defined as the size ratio at 84.2% on the cumulative frequency curve to the median diameter. This assumes that the distribution of particle sizes is lognormal. A monodisperse, i.e. ideal aerosol, has a GSD of 1, although in practice an aerosol with a GSD of <1.22 is described as monodisperse while those aerosols with a GSD >1.22 are referred to as poly dispersed or heterodispersed. 

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. 

Fig. 7. Structural log of core entering a fault damage zone. The high structural frequency, faults and fractures concentrated at the base of the core, are arranged in clusters which define steps in the cumulative frequency curves.

The cumulative frequency table gives the number of observations less than a given value. Probabilities can be read from the cumulative frequency curve (Figure 131). For example, to find the probability that a value will be less than 85, one should read the curve at the point x = 85 and read across to the value 0.74 on the y axis. 

Probability graph paper is used in the analysis of cumulative frequency curves for example, graph paper can be used as a rough test of whether the arithmetic or the logarithmic scale best approximates a normal distribution. The scale, arithmetic or... 

Figure 12. Results of particle size analysis of a sample from West Basin Lake. Australia. A Histogram B Cumulative frequence curve plotted using arithmetic- scales C Cumulative frequency curve plotted using probability-0 scales.

Measures of Central Tendency Folk s Graphic Mean Inman Mean Median Mode = (di 6 + < 5U + S4)/3 Mg = 050 Ma = Most frequently occurring panicle size. On a histogram it is the midpoint of the most abundant class interval on a cumulative frequency curve it is the size conespontiing to the steepest part of the curve. 

The data in Table 3 are expressed in a cumulative frequency curve as shown in Fig. 8. ISO 6989 is somewhat similar to ASTM D 5103 but more realistically measures 500 fibers and also permits the use of the WIRA fiber length machine. The modal length (i.e., the central length of the most numerous class length) and the mean length are calculated. Either a histogram or a cumulative frequency diagram can be prepared (see also ASTM D 3661, below). 

Figure 8 Cumulative frequency curve for fiber length.

Figure 1 The normal (Gaussian) distribution represented as (A) a frequency curve and (B) a cumulative frequency curve.

Water quality monitoring consists of frequent analysis of the main constituents. The required data input consists of (1) mean composition of the influent (2) mean composition of native groundwater in each layer of the target aquifer (3) native geochemistry of each layer of the target aquifer (4) the cumulative frequency curve of detention times in each model layer or flow path as derived from either separately run hydrological model or tracer breakthrough data and (5) specific information derived from the mass balance of the water phase (the reactions that are needed, how O2 and NO3" distribute over the various redox reactions, etc.). 

Critical values for one-tailed and two-tailed tests at P=0.05. The appropriate value is compared with the maximum difference between the experimental and theoretical cumulative frequency curves. 

As has been emphasized in this chapter, many statistical tests assume that the data used are drawn from a normal population. One method of testing this assumption, using the chi-squared test, was mentioned in the previous section. Unfortunately, this method can only be used if there are 50 or more data points. It is common in experimental work to have only a small set of data. A simple visual way of seeing whether a set of data is consistent with the assumption of normality is to plot a cumulative frequency curve on special graph paper known as normal probability paper. This method is most easily explained by means of an example. 

Figure 3.3 The cumulative frequency curve for a normal distribution.

---