Class interval

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

The distribution curves may be regarded as histograms in which the class intervals (see p. 26) are indefinitely narrow and in which the size distribution follows the normal or log-normal law exactly. The distribution curves constructed from experimental data will deviate more or less widely from the ideal form, partly because the number of particles in the sample is necessarily severely limited, and partly because the postulated distribution... 

Determine the width of each class interval (column)... 

Put class interval in the X-axis Put frequency scale on the Y-axis... 

The asymmetry of peak shape is preserved in anthraxolite heated to 1200°C. showing that turbostratic disorder persists in spite of a general enhancement of ordering. The band is also sharper and narrower. This may be interpreted to mean either that fewer class intervals are represented in the crystallite size distribution or that increased ordering of aromatic lamellae has reached the point where graphite (hid) planes are more common. Diffraction peaks of both (100) and (101) fall with the 2-A. band. 

Fig. 37.3 Frequency distribution of weights for a range of different soil types (sample size 24085) the size class interval is 2g.

A useful function where you have large numbers of data allows you to create frequency distributions using pre-defined class intervals. 

Histogram A graphical representation of a frequency distribution. Rectangles are used. Their heights represent the frequencies, and their widths represent the class interval. 

Figure 8. Variation of median arsenic concentration for well waters within a given iron concentration class interval...

As a further step, one can graph the information in the frequency table. One way of doing this would be to plot the frequency midpoint of the class interval. The solid line connecting the points of Figure 129 forms a frequency polygon. 

Figure 129. Pollution concentration (midpoint of class interval) frequency polygon.

Class Interval Tally Frequency Cumulative Frequency... 

Tables are the ideal method for describing qualitative data sets, and several examples are shown in Section 7.3. Such tables present the actual numbers or frequencies in each of the categories, often with percentages. Tables of this sort are also useful for ordinal data if there are relatively few categories or scale values. With quantitative data, or ordinal data in which there are many categories or scale values, data can be grouped into class intervals so that a tabular presentation can be made.

Sturges, H.A. The choice of a class interval. Journal of the American Statistical Association 1926 21 65-66. 

When = 1 the particulates are distributed equally in each of the class intervals. We have seen, however, that many aqueous particulates exhibit values of p greater than two. At this value of p, computations using Equation 3 show that the total number concentration is dominated by fine particulates. Data in Figure 1 confirm this analysis. 

For a hypothetical particulate suspension with sizes in the measurable range of 0.5 ju,m-100 fim and a 8 = 3, the relative contribution of given size classes to the total number, area, and volume concentration is shown in Figure 2, computed using Equations 3 and 4 with the number of class intervals i = 12. Particles smaller than 3 /xm dominate the total number concentration for this distribution, while the volume concentration is controlled by particulates larger than 3 /xm. Each size class contributes equally to the total surface area of the distribution, but for the size limits selected, particulates larger than 3/xm contribute approximately 75% of the total area. 

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. 

ABSTRACT DDFPM has been developed as an alternative for Monte Carlo in the assessment of structural reliability in probabilistic calculations (Marek et al. 1995). Input random quantities (such as the load, geometry, material properties, or imperfections) are expressed as histograms in the calculations. In the probabilistic calculations, all input random variables are combined with each other. The munber of possible combinations is equal to the product of classes (intervals) of all input variables. With rather many input random variables, the number of combination is very high. Only a small portion of possible combinations results, typically, in failures. When DDFPM is used, the calculation takes too much time, because combinations are taken into account that does not contribute to the failure. Efforts to reduce the number of calculation operations have resulted into the development of algorithms that provide the munerical solution of the integral that defines formally the failure probability with rather many random variables ... 

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