Line graphs, purpose

The central purpose of a graph is to present, summarize, and/or highlight trends in data or sets of data. Graphs of various types (e.g., scatter plots, contour plots, two- and three-dimensional line graphs, and bar graphs) are used for different purposes thus, authors must match their purpose with the appropriate type of graph. 

Scatter Diagram A scatter diagram is a basic tool to identify the potential relationship between two variables. Scatter diagrams are similar to line graphs in that they use horizontal and vertical axes to plot data points. Flowever, they have a very specific purpose. Scatter diagrams show how much one variable is affected by another. The relationship between two variables is called their correlation. The... 

Solution The classic way of fitting these data is to plot ln( /7 ) versus T" and to extract and Tact from the slope and intercept of the resulting (nearly) straight line. Special graph paper with a logarithmic j-axis and a l/T A-axis was made for this purpose. The currently preferred method is to use nonlinear regression to fit the data. The object is to find values for kQ and Tact that minimize the sum-of-squares ... 

However, large errors can arise when the regression is attempted in improper coordinates, or, what is equivalent, if a line is drawn approximately through a given set of points. Unobjectionable statistical means are now available (Section IV.) to decide whether the relationship holds and to obtain the value of j3. The purpose of various graphs is only to represent objectively the results obtained. 

The middle section of a curve may cross some important points as previously marked on the graph. Make sure that the curve does, in fact, cross these points rather than just come close to them or you lose the purpose of marking them on in the first place. Always try to think what the relationship between the two variables is. Is it a straight line, an exponential or otherwise and is your curve representing this accurately ... 

Figure 5.10 The fraction of dose dissolved ipi as a function of generations i, where the solid line represents the fittings of (5.24) to danazol data [126]. Symbols represent experimental points transformed to the discrete time scale for graphing and fitting purposes assigning one generation equal to 15 minutes. Key (% sodium lauryl sulfate in water as dissolution medium) 1.0 0.75 A 0.50 T 0.25 0.10.

If the functional relationship between one variable and another is linear, a straight-line plot would be obtained on arithmetic-coordinate graph paper. If the relationship approaches a linear one, the best method of fitting the data to a linear model would be through the method of least squares. The resulting linear equation (or line) would have the properties of lying as close as possible to the data. For statistical purposes, close and/or best fit is defined as that linear equation or line for which the sum of the squared vertical distances between the data (values of Y or independent variable) and line is minimized. These distances are called residuals. This approach is employed in the solution below. 

The heights of the bars or columns usually represent the mean values for the various groups, and the T-shaped extension denotes the standard deviation (SD), or more commonly, the standard error of the mean (discussed in more detail in Section 7.3.2.3). Especially if the standard error of the mean is presented, this type of graph tells us very litde about the data - the only descriptive statistic is the mean. In contrast, consider the box and whisker plot (Figure 7.2) which was first presented in Tukey s book Exploratory Data Analysis. The ends of the whiskers are the maximum and minimum values. The horizontal line within the central box is the median, fhe value above and below which 50% of the individual values lie. The upper limit of the box is the upper or third quartile, the value above which 25% and below which 75% of fhe individual values lie. Finally, the lower limit of the box is the lower or first quartile, the values above which 75% and below which 25% of individual values lie. For descriptive purposes this graphical presentation is very informative in giving information about the distribution of the data. 

Figures 3.10c and 3.10d are representations of the same ternary system in terms of weight fraction and weight ratios of the solute. In Fig. 3.10d the ratio of coordinates for each point on the curve is a distribution coefficient K p.= YfiXf. If K p were a constant, independent of concentration, the curve would be a straight line. In addition to their other uses, x-y or X-Y curves can be used to obtain interpolate tie lines, since only a limited number of tie lines can be shown on triangular graphs. Because of this, x-y ot X-Y diagrams are often referred to as distribution diagrams. Numerous other methods for correlating tie-line data for interpolation and extrapolation purposes exist.

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