Representing numerical information so that it can be easily grasped is impor-tant to anyone using numbers on the job. A random list of numbers may hide important trends, whereas a table or a graph based on these numbers might show these trends clearly. For example, compare the following ways to pres-ent data. First, read the spres-entence, “At 1:00 p.m., the pressure reading on a gas pipeline was 675 psig; at 2:00 p.m., 670 psig; at 3:00 p.m., 665 psig; at 4:00 p.m., 655 psig; and at 5:00 p.m., 650 psig.” Now, look at the same information in tabular form (fig. 3.6) and on a graph (fig. 3.7). Carefully reading the sentence and the table reveals that the pressure is falling steadily, but the graph shows the trend at a glance.
Using Reference Tables
Large amounts of information can be presented in a relatively small amount of space when presented in a table. Moreover, a table can show the data in a way that makes it easy to understand.
GAS PIPELINE PRESSURE
Reference information can be conveniently presented in tabular form. Also, knowing how to read and use tables can be important when solving mathematical problems. For example, figure 3.8 is a table that gives the squares, cubes, square roots, and cube roots for numbers from 1 to 50. You can use the table to find any of these properties for a given number, or for finding the number if one of the properties is known.
Example Problem: Find the cube of 47, using the table in figure 3.8.
Solution: Find the number 47 in the left-hand column labeled No. Then, move horizontally across the table to the cube column. The answer—103,823—is found where the two columns intersect.
You can also use the table to find an unknown number. For example, you can find the value for a in the equation
a3 = 39,304.
Locate the number 39,304 in the cube column, then read the answer, which is 34, across from it in the No. column. Thus, 343, is 39,304.
Interpolation
Sometimes, even a comprehensive table does not show the specific information needed for working problems. Fortunately, when you need a value that falls be-tween the figures given in a table, you can interpolate to come up with the value.
To interpolate means to insert or estimate values between two known values. For example, let’s say you need the square root of 800. Looking at figure 3.8, note that the table does not give this information. However, it does show that the square root of 784 is 28 and the square root of 841 is 29. So, the square root of 800 is between 28 and 29. To interpolate this information, first find the difference in the two numbers given:
841 – 784 = 57.
Then, find the difference in the number desired and the smaller known number:
800 – 784 = 16.
The number 16 shows that the square root of 800 is ¹⁶⁄57, or 0.28, more than the square root of 784, which is 28. So,
28 + 0.28 = 28.28.
Therefore, 28.28 is the square root of 800 (or 28.282 = 800). If you have a cal-culator with the square root function, use it to check the answer. Depending on how many places your calculator carries the solution to, the answer should be something like 28.28427125, which is very close to 28.28. (Actually, 28.282 = 799.7584, which, for most purposes, is close enough.)
Constructing Tables
Most tables are constructed to present numerical or statistical information in the form of percentages, amounts of money, number of occurrences, and the like. In any table, one set of information represents the primary focus of the table, and a second set of data represents varying conditions or other related variables. 47 2,209 103,823 6.86 3.61 48 2,304 110,592 6.93 3.63 49 2,401 117,649 7.00 3.66 50 2,500 125,000 7.07 3.68
Figure 3.8 Reference table
72 NUMBER RELATIONS
A simple table is constructed with the primary set of variables (known as boxheads) across the top of the table and the second set of variables (known as the stub) down the left side of the table (fig. 3.9). The body of the table presents the desired statistics. The table’s title is important because it affects the amount of information that the boxhead must present. Boxheads or stubs also need units of measure to qualify them, such as percent, dollars, inches, metres, degrees Fahrenheit, and so forth.
Example Problem: According to projections published by the Pipeline & Gas Journal in 2000, the number of miles of main gas distribution utility piping to be installed between 2000 and 2004 were as follows: 2000—14,000; 2001—14,200;
2002—14,210; 2003—14,200; and 2004—15,100. The number of miles of service lines during the same period was projected to be the following: 2000—15,000;
2001—15,600; 2002—15,500; 2003—15,900; and 2004—15,900. Prepare a table that presents these statistics as well as the total number of miles of piping installed for each year.
Solution: Pick out the primary focal point of the table and use these points as box heads: Year, Main, Service, and Total. For these heads to be meaningful, a unit of measurement (miles) should be added to Main, Service, and Total. Then, the years become the stub, and the statistics become the body as shown in figure 3.10.
Plotting Graphs
A graph is a diagram that indicates relationships between two or more variables.
Types of graphs include bar graphs, line graphs, and circle graphs, which are also called pie charts. An axis is one of usually two straight lines on a line graph—that is, a line graph usually has two axes. Reference points called coordinates are placed on the axes (fig. 3.11). Axes are commonly drawn perpendicular to each other. The horizontal axis is the abscissa, or X axis; the vertical axis is the ordinate, or Y axis.
The location of any point on a line graph is given as its perpendicular distance from the two axes. For example, point A in figure 3.11 is located 5 units from the X axis and 4 units from the Y axis, or 4 across and 5 up. These dimensions are often referred to as coordinates; in this case, the coordinates of point A are 4, 5.
PROJECTED GAS UTILITY Source: “Gas Distribution Utility Piping Market Statistics,” Pipeline & Gas Journal,2000.
Figure 3.10 Table data
Figure 3.11 Pair of reference axes
8
Tables and Graphs 73
Arrange Data in Tabular Form
Suppose that you wish to plot a graph for the number of rigs that were running at XYZ Drilling Company for the months January through June. Before a graph is plotted, arrange the data in tabular form. Arrange the data to be plotted on the X axis in one column—the months in this case—and the data for the Y axis—the number of rigs in this case—in a parallel column with corresponding items opposite each other (fig. 3.12).
Choose Scale and Plot Points
The scale chosen for a graph depends on the amount of space available and on the type of data to be shown. Scales are normally chosen so that the maximum value along the Y axis uses about 80 to 90 percent of the available space, and the X axis completely fills the available space. Standard graph paper is often used as a base and it determines the size and scale of the graph. Using the data arranged in tabular form and the scale, the point can be plotted (fig. 3.13).
The next step is to decide the type of graph to show the plotted information.
XYZ DRILLING COMPANY
Figure 3.13 Points plotted for a graph
Figure 3.12 Information arranged in tabular form
As previously mentioned, three types of graphs are the bar graph, the line graph, and the circle graph. Line graphs may have broken lines or curved lines, depend-ing on the data used. Each type of graph offers information at a glance, such as a trend, a comparison, or a curve.
Bar Graphs
The bar graph is a good way to present data that represents a series of obser-vations made at periodic intervals. For example, the bar graph in figure 3.14 depicts the tabular information given in figure 3.12 using the points plotted in figure 3.13. The graph shows the relationship between the number of rigs running in January, February, March, and so on. A bar graph is a good way
Figure 3.14 Bar graph showing tabular data given in figure 3.12
74 NUMBER RELATIONS
to present unrelated facts—that is, facts where one fact is not dependent on the other.
Line Graphs
The line graph can also be used to depict unrelated facts (fig. 3.15); however, it is more commonly used to plot two related variables, such as temperature and time or pressure and time. If the information represents irregular variations, it is plotted by a broken line joining the known points (fig. 3.16; also see fig. 3.15). If the information represents variables depending directly on each other, a curved line may be used to join the plotted data (fig. 3.17).
Figure 3.15 Line graph showing tabular data given in figure
3.12 Figure 3.16 Broken-line graph
Figure 3.17 Curved-line graph
170
Tables and Graphs 75
Curved-line graphs are also used to show relationships in general—without specific statistics. For example, figure 3.18 is a graph that plots the vapor pressure for a pure substance at increasing pressures and temperatures.
(180OIL°)
WATERFREE (90°) EMULSION
(72°)
SED.(18°)
Figure 3.19 Circle graph or pie chart
Figure 3.18 Curved graph showing a general relationship
PRESSURE
TEMPERATURE VAPOR PRESSURE CURVE
FOR A PURE SUBSTANCE
LIQUID VAPOR PC
TC C
PC—Critical Pressure C—Critical Point TC—Critical Temperature
Circle Graphs
A circle graph, also called a pie chart, is often used to compare the various parts of a whole to each other and to the whole. With 360° equaling the whole, the circle is divided so that each 1% of the whole is represented by 3.6°. To construct a circle graph, it is necessary to find the size of the angle for each part. Find angle size by using the formula
part
––––– × 360° = angle.whole
When each angle is determined, it is plotted with a protractor and lines are drawn inside a circle.
Example Problem: A well produces 180 barrels of fluid a day. Ninety barrels are oil, 45 barrels are free water, 36 barrels are emulsion, and 9 barrels are sediment.
Make a circle graph, or pie chart, to illustrate this distribution.
Solution: Calculate the angles of each “slice of pie” using the formula:
part ÷ whole × 360° = angle.
90 ÷ 180 × 360° = 180° (oil) 45 ÷ 180 × 360° = 90° (free water) 36 ÷ 180 × 360° = 72° (emulsion)
9 ÷ 180 × 360° = 18° (sediment) Then plot the graph as shown in figure 3.19.
76 NUMBER RELATIONS
Practice Problems
1. A worker earned $286.50 one week, of which $60 was spent on rent, $70.80 on food, $7 on laundry, $32.50 on gasoline, $63.40 on incidentals, and the rest was saved. Plot a circle graph showing how the worker’s weekly salary was distributed. (Round off degrees to whole numbers before plotting.)
2. Using the graph of OPEC production and demand shown below as a reference, give approximate answers to the following questions.
a. In what month and year did the excess capacity cease, creating a supply shortage?
_____________________________________________________
b. From 1981 to 1982, did the demand for OPEC oil increase or decrease?
_____________________________________________________
YEAR
1976 1978 1980 1982 1984 1986
MILLION BBL/DAY
60
50
40
30
20
10
0
DEMAND OPEC OILFOR SUPPLY SHORTFALL PRODUCTIONOPEC
CAPACITY EXCESS CAPACITY
PROJECTION OF OPEC PRODUCTION AND DEMAND
c. How much was the excess capacity of OPEC oil production at the beginning of 1976?
_____________________________________________________
d. In 1984, what was the average daily OPEC capacity to produce oil?
_____________________________________________________
e. In what year did OPEC production capacity begin to gradually level out?
_____________________________________________________
3. The pressure of a certain volume of gas at 50°F is 1,000 psig. At 60°F the pressure increases to 1,020 psig; at 70°F to 1,040 psig; at 80°F to 1,060 psig; and at 100°F to 1,100 psig. Plot a smooth curve from this data on a line graph.
4. The total miles of gas pipe installed by type between 1995 and 2000 are to be put into a table. Statistics for plastic pipe installed are 1995—15,985;
1996—11,640; 1997—15,991; 1998—14,576; 1999—18,826; and 2000—
18,912. Similar statistics for steel pipe are 1995—12,663; 1996—9,718;
1997—6,157; 1998—7,212; 1999—8,145; and 2000—8,334. Construct a table giving this information.
Tables and Graphs 77
78 NUMBER RELATIONS
5. Total world crude production varied from year to year between 1976 and 1980, but offshore crude production showed a steady growth. Using the statistics given in the table to the left, construct a bar graph that shows total production and offshore production. (Note: Shade a part of each bar to show the offshore production.)
AVERAGE DAILY WORLD CRUDE PRODUCTION
(in millions of bbl) YEAR OFFSHORE TOTAL
1976 8.5 58.5
1977 10.0 56.5
1978 11.0 60.0
1979 12.0 62.5
1980 13.5 59.0