Graphical Presentation of Data

1. Graphical Presentation of Data • Descriptive Statistics can be divided into two subject areas • Graphical methods, and • Numerical methods. • In this section, we introduce several graphical techniques useful for presenting and summarising data, and in section 1-3 and 1-4 we will present some numerical summary methods. • Notationally, we will let n represent the number of observations in a data set, and the observations themselves will be represented by a subscripted variable, say x1, x2… xn. • Thus, n=5 values of compressive strength of concrete measured by a civil engineer could be represented by X1= 2310, X2= 2325, X3= 2315, X4= 2340, and X5= 2335 (units are psi)

2. The Dot Diagram • Montgomery (1991) describes an experiment in which an engineer has added polymer latex to portland cement mortar to determine its effects on tension bond strength (in kgf/cm2). • The data that result from this experiment are 16.85, 16.40, 17.21, 16.35, 16.52, 17.04, 16.96, 17.15, 16.59 and 16.57 • These data are plotted as a dot diagram in Fig. 1-1.

3. The Dot Diagram Continued • The dot diagram is a very useful plot for displaying a small body of data, say, up to about 20 observations. • This plot allows us to quickly and easily see the location or central tendency in the data and the spread or variability. • For example, note that the middle of the data is very close to 16.8 and that the values of tension bond strength fall in the interval from about 16.3 to 17.2 kgf/ cm2.

4. The Dot Diagram Continued • Dot diagrams are often helpful in comparing two or more data sets. • For example, 10 measurements of tension bond strength in unmodified portland cement mortar are 17.50, 17.63, 18.25, 18.00, 17.86, 17.75, 18.22, 17.90, 17.96 and 18.15. • Note that the dot diagram immediately reveals that modified mortar seems to result in lower tension bond strength, but that the inherent variability within both groups of measurements is about the same.

5. Selecting the Suitable Graphical Display • When the number of observations is small, it is often difficult to identify any specific pattern of variation, however, the dot diagram will frequently be helpful and may provide information about unusual features in the data. • When the number of observations is moderately large, other graphical displays may be more useful.

6. Selecting the Suitable Graphical Display Continued • For example, consider the data in table 1-1, these data are the comprehensive strengths in psi of 80 specimens of a new aluminium- lithium alloy undergoing evaluation as a possible material for aircraft structural elements. • These data were recorded in the order of testing, and in this format they do not convey much information about compressive strength. • Questions such as what percent of the specimens fail below 120 psi? Are not easy to answer. • Because there are many observations, constructing a dot diagram of these data would be relatively inefficient, more effective displays are available for large data sets.

7. Stem-and-Leaf Diagram • A stem-and-leaf diagram is a good way to obtain an informative visual display of a data set x1, x2… xn, where each number xi consists of at least two digits. • To construct a stem-and-leaf diagram, we divide each number xi into two parts: • a stem, consisting of one or more of the leading digits, and • a leaf, consisting of the remaining digits.

8. Stem-and-Leaf Diagram Continued • To illustrate, if the data consists of percent defective information between 0 and 100 on lots of semiconductor wafers, then we can divide the value 76 into the stem 7 and the leaf 6. • In general, we should choose relatively few stems in comparison with the number of observations. • It is usually best to choose between 5 and 20 stems. • Once a set of stems has been chosen, they are listed along the left-hand margin of the diagram. • Beside each stem all leaves corresponding to the observed data values are listed in the order in which they are encountered.

9. Example 1-1 • To illustrate the construction of a stem-and-leaf diagram, consider the alloy compressive strength data in table 1-1. We will select as stem values the numbers 7, 8, 9… 24. • The resulting stem-and-leaf diagram is presented in Fig. 1-3. The last column in the diagram is a frequency count of the number of leaves associated with each stem. • Inspection of this display immediately reveals that most of the compressive strengths lie between 110 and 2000 psi and that a central value is somewhere between 150 and 160 psi. • Furthermore, the strengths are distributed symmetrically about the central value. • The stem-and-leaf diagram enables us to determine quickly some important features of the data that were not immediately obvious in the original display in Table 1-1.

10. Modifying the Original Stems • In some data sets, it may be desirable to provide more classes or stems. • One way to do this would be to modify the original stems as follows: • Divide the stem 5 (say) into two new stems, 5L and 5U. • The stem 5L has leaves 0, 1, 2, 3, and 4; • and stem 5U has leaves 5, 6, 7, 8 and 9. • This will double the number of original stems. • We could increase the number of original stems by four by defining five new stems: • 5z with leaves 0 and 1, • 5t (for twos and three) with leaves 2 and 3, • 5f (for fours and five's) with leaves 4 and 5, • 5s (for six and seven) with leaves 6 and 7, and • 5e with leaves 8 and 9.