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Cumulative Frequency Distribution Cumulative Frequency Distribution The total frequency of all classes less than the upper class boundary of a given class is called the cumulative frequency of that class. “A table showing the cumulative frequencies is called a cumulative frequency distribution�. There are two types of cumulative frequency distributions. Less than cumulative frequency distribution : It is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The cumulate is started from the lowest to the highest size. More than cumulative frequency distribution: It is obtained by finding the cumulate total of frequencies starting from the highest to the lowest class. The less than cumulative frequency distribution and more than cumulative frequency distribution for the frequency distribution given below are: A frequency distribution is one of the most common graphical tools used to describe a single population. It is a tabulation of the frequencies of each value (or range of values). There are a wide variety of ways to illustrate frequency distributions, including histograms, relative frequency histograms, density histograms, and cumulative frequency distributions. Know More About Correlation Examples

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Histograms show the frequency of elements that occur within a certain range of values, while cumulative distributions show the frequency of elements that occur below a certain value. Relative Frequency Histogram Relative frequency defined as the fraction of times the value occurs, or the freuqency of value(s) รท number of observations in the set. Relative frequencies usually of more interest than the absolute frequencies. Relative frequency histogram constructed by assigning the relative frequencies as heights of the rectangles. Sum of all relative frequencies in a dataset is 1. Density Histogram Similar to frequency histogram except heights of rectangles are calculated by dividing relative frequency by class width. Resulting rectangle heights called densities, vertical scale called density scale. Noteworthy property: (class width * density) = relative frequency. Total area of all rectangles equals 1. Histogram Shapes Unimodal: Rises to single peak, then declines. Bimodal: Has two distinct peaks. Multimodal: More than two peaks. Discriptions of skew may also be applied to histograms (see Measures of Central Tendency section.)

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Line Graph Definition Line Graph Definition In graph theory, the line graph L(G) of undirected graph G is another graph L(G) that represents the adjacencies between edges of G. The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this. Other terms used for the line graph include edge graph,the theta-obrazom,the covering graph,the derivative,the edge-to-vertex dual,the interchange graph, the adjoint graph,the conjugate, the derived graph,and the representative graph. One of the earliest and most important theorems about line graphs is due to Hassler Whitney (1932), who proved that with one exceptional case the structure of G can be recovered completely from its line graph. In other words, with that one exception, the entire graph can be deduced from knowing the adjacencies of edges ("lines"). A source of examples from geometry are the line graphs of the graphs of simple polyhedra. Taking the line graph of the graph of the tetrahedron one gets the graph of the octahedron; from the graph of the cube one gets the graph of a cuboctahedron; from the graph of the dodecahedron one gets the graph of the icosidodecahedron, etc.

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Geometrically, the operation consists in cutting each vertex of the polyhedron with a plane cutting all edges adjacent to the vertex at their midpoints; it is sometimes named rectification. If a polyhedron is not simple (it has more than three edges at a vertex) the line graph will be nonplanar, with a clique replacing each high-degree vertex. The medial graph is a variant of the line graph of a planar graph, in which two vertices of the medial graph are adjacent if and only if the corresponding two edges are consecutive on some face of the planar graph. For simple polyhedra, the medial graph and the line graph coincide, but for non-simple graphs the medial graph remains planar. Thus, the medial graphs of the cube and octahedron are both isomorphic to the graph of the cuboctahedron, and the medial graphs of the dodecahedron and icosahedron are both isomorphic to the graph of the icosidodecahedron. By the result of Whitney (1932), if G is not a triangle, there can be only one partition of this type. If such a partition exists, we can recover the original graph for which G is a line graph, by creating a vertex for each clique, and connecting two cliques by an edge whenever G contains a vertex belonging to both cliques. Therefore, except for the case of and , if the line graphs of two connected graphs are isomorphic then the graphs are isomorphic. Roussopoulos (1973) used this observation as the basis for a linear time algorithm for recognizing line graphs and reconstructing their original graphs

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Cumulative Frequency Distribution