Data Visualization/Graphical Analysis

A qualitative idea of the central tendency and the dispersion of the sample data can be obtained by representing it graphically. Typically, graphical visualization precedes quantitative statistical analysis. Figure 1 shows a scatter plot of the data, i.e., the data points x₁, x₂, ..., x_n plotted on a 2d graph against their indices. Some of the key statistical features of the data revealed by the scatter plot are: (a). a qualitative idea of the nature of variability, in this case it looks like a random scatter around an average melting point of approximately 320 and (b). the maximum and minimum values of the melting point which are 335 and 305 respectively.

A qualitative idea of how the data are distributed between the maximum and minimum values can be obtained by constructing a frequency plot, typically presented as a histogram. Here, we plot the number of observations of x_i, say f_i, vs. x_i. A frequency histogram for the melting point data above is given in Figure 2. From the frequency plot, we can see that the melting point of 320 occurs most frequently, i.e., f (320) = 5. The observation with the largest frequency is called the mode of the sample.

We can also tabulate the number of observation within a given interval and plot the frequency data thus obtained against the mid-points of the corresponding intervals. For instance, we can create 7 equal intervals: [302.5,307.5), [307.5,312.5), [312.5,317.5), [317.5,322.5), [322.5,327.5), [327.5,332.5), [332.5,337.5). The interval [a, b) contains the data greater than or equal to a and less than b. In Figure 3, you can see a plot of the number of observations of the melting point measurements within each one of these 7 intervals vs. the midpoint of the corresponding interval. As Figure 3 reveals, the distribution is approximately symmetric about a melting point value of 320.