GRAPHING

Written by: Bette Kreuz

Edited by: Science Learning Center Staff

The Graphing module consists of written material, examples of graphs, and a posttest (taken in the SLC).

The objectives are to:

1) select suitable graph paper

2) identify the dependent and the independent variables

3) scale and label the axes

4) title the graph

5) plot the data points

6) draw a smooth curve

Directions

This module consists of a written material and examples of graphs. As you read through the material, you will be instructed to refer to the examples of data and graphs that illustrate the various aspects of graphing covered by this module. At certain points you will be asked to stop and formulate answers to questions. When you feel you have the answers requested, simply continue with the module.

When you have completed the module, ask for a Post-test consisting of a set of data and the materials needed to construct a graph of the data. Construct the graph following the steps presented in the module and then turn the graph in to be checked. Also return any other materials that have been given to you. If you pass the Post-test, make sure your name is recorded in the SLC database. If you do not pass it, you may review the module and retake the test as many times as needed.

Introduction

A GRAPH CAN BE USED TO:

1) COMMUNICATE INFORMATION

2) DERIVE QUANTITIES NOT DIRECTLY MEASURED

AND A GRAPH SHOULD DO THIS:

1) CLEARLY

2) COMPLETELY

3) ACCURATELY

Graphing is an important means of communicating information.

Relationships between variables (quantities which can have more than one numerical value) or sets of data may not be readily evident when displayed in a table or contained in the body of a report, but often become clearly evident when an appropriate graph of the variables or data is made. In addition, calculations such as the slope of the graph of the y-intercept can yield further information which may not be easy to measure experimentally. For a graph to provide such information, the graph must clearly, completely and accurately present the information in a simple and easy to follow manner. Although other means exist to display such information, graphing is one of the most commonly used methods. In order for a graph to meet the above criteria, standard rules for the construction of graphs must be observed.

Purpose

The purpose of this module is to present these standard rules for the construction of a graph in a series of simple to follow steps that should be applicable to the graphing of most types of data that you will encounter. This will be done by taking experimental data and following though the steps needed to construct a graph of the data. At the end of the module you will be given a set of data and asked to construct an appropriate graph of the data in order to ascertain whether or not you need to review any of the steps presented.

Basic Principles

TABLE 1: Volume Variation of One Mole of an Ideal Gas at One Atmosphere as a Function of Absolute Temperature

Volume (liter)

Temperature (° K)

8.2

100

16.4

200

24.6

300

32.8

400

41.0

500

49.3

600

57.5

700

Presented here is a set of data taken by a student during the course of an experiment. You will note that the data in the table have been arranged in an orderly fashion and the table is clearly labeled as to the measured quantities (volume andtemperature of a gas) and the units of measurement (liters and degrees Kelvin or K).

This set of data has been graphed in two ways as shown in Figure 1 and Figure 2.

In Figure 1 the graph was constructed following standard format, and it clearly displays the relationship between the two quantities measured.

In Figure 2, another graph of the data is shown. This graph, unlike the first one, does not display the information clearly.

For example: What temperature scale was used? What does the vertical scale represent? Where are the actual data points located? Is the relation non-linear, etc.? The errors made on this graph that contribute to its lack of clarity would not have been made if the standard rules for graphing had been followed.

Two basic principles can serve as a guide in constructing a graph. These are:

1) The graph should cover most of the paper rather than being squeezed into a tiny area. If the points on a graph are very close together, it loses much of its significance. On the other hand, the graph should not be spread out so far that some of the points are off the paper.

2) The graph should be clear, easy to read and easy to construct. In particular, it should be possible to determine the coordinates (x and y values) of any of the points on the graph with a minimum of effort and to see immediately what they represent (e.g. temperature in ° K, solubility in grams per liter).

Although variations exist, if the following steps are followed in constructing any graph, your graph will be acceptable.)

STEPS IN GRAPHING

STEP 1: Select the graph paper.

STEP 2: Identify the dependent and the independent variables.

STEP 3: Scale the axes.

STEP 4: Assign values to the origin of the graph.

STEP 5: Number each major division along each axis.

STEP 6: Complete labeling of both axes.

STEP 7: Title the graph.

STEP 8: Plot the data points.

STEP 9: Draw a smooth curve.

Now we will put the rules into practice working with a data set for Figure 3, Solubility of KNO3 as a Function of Temperature.

STEP 1: Select the graph paper.

First, paper with ruled lines must always be used in constructing a graph. Graphing should not be done on plain or notebook paper. Accuracy in plotting and reading the points is lost if such paper is used. Many types of commercial graph paper are available: semi-logarithmic, full logarithmic, linear grids and many others. Some of the types available are shown here. The type you should use depends on the data at hand.

Graph Paper Example Use?
Applicable for present purposes
Applicable for present purposes
Applicable for present purposes
Not Applicable here; does not allow for plotting points with the accuracy allowed by most experimental procedures
Not Applicable here; this is semi-logarithmic paper and will be described in another learning module
Not Applicable here; this is logarithmic paper and will be described in another learning module

STEP 2: Identify the dependent and the independent variables.

Table 2: Solubility of KNO3 as a Function of Temperature

Solubility of KNO3 (g/100 g water)

Temperature (° C)

13

0

21

10

32

20

46

30

64

40

86

50

110

60

138

70

170

80

208

90

248

100

This set of data was taken in an experiment in which the students measured the solubility of the salt potassium nitrate at the temperature listed.

The temperatures were controlled by a hot water bath. In this set of data the temperature was the factor that the students controlled. Therefore, it is the independent variable, and as such should be placed on the horizontal or x-axis of the graph.

Solubility of potassium nitrate was measured as a function of temperature. Therefore, solubility is the dependent variable and as such is placed on the vertical or y-axis of the graph.

The Figure below highlights placement of variables along axes for Figure 3.

Test your understanding.

Consider these examples which describe two experiments in which the students would eventually graph the data. Read each example and determine what you believe to be the dependent and the independent variable in each. Stop now, read the examples, and when you have formulated your answers, continue with the module.

Example 1:

In an experiment in which students were studying the sensitivity of green plants to sunlight, they extracted the pigment chlorophyll from the plants. They then measured how this pigment absorbed light at certain wavelengths in the visible region of the spectrum. Which is the dependent variable, wavelength of light of absorption of light? Which is the independent variable?

Example 2:

During a titration an electrode to monitor pH was placed in the titration flask and the pH of the solution was read off after each milliliter of sodium hydroxide that was added to the flask. The two variables are pH and volume of sodium hydroxide in milliliters. Which is the dependent variable and which is the independent variable?

Answers:

Example 1: the independent variable is wavelength of light (the controlled variable) and the dependent variable is absorption of light.

Example 2: The independent variable is the volume of sodium hydroxide (the controlled variable) and the dependent variable is the pH since it was measured on the addition of controlled amounts of sodium hydroxide to the titration flask.

STEP 3: Scale the axes.

The purpose of this step is to decide how many of the major divisions of the graph paper to use for each axis and how many units of measure the divisions along each axis will represent. Your decision here will depend on the particular set of data that you will be graphing but the following general method should be followed. We will use the solubility data given in Table 2 above.

Let's consider the horizontal or x-axis first. This will be used for the independent variable, the temperature values, as shown in Table 2. The temperature values run from 0° to 100° or a range of 100° . If we utilize 10 of the major divisions on the graph paper then each of the major divisions will correspond to:

100° /10 = 10°

and each of the minor divisions (there are 5 per major division) will correspond to:

10° /5 =2°

Usually it is convenient to set up a scale where each of the major divisions is a multiple of 2, 5, 10, 20 etc. since this makes reading the minor divisions easier and thus makes the whole graphing process simpler. These numbers are convenient (multiples of 10, 5, 2 or 1 usually are) and allow for plotting the temperatures with the precision shown in the table.

Next consider scaling the vertical or y-axis. The dependent variable, solubility values, run from 13 g/100 g water to 248 g/100 g water or a range of 235 g/100 g water. Again we want to scale the axis so that we can plot points to the precision indicated by the data, here to the neared 1g/100 g water.

Note: It is not necessary to start either axis at zero, as long as the starting point is clearly indicated.

If we start the vertical axis at 10 and end at 250, the scale has a range of 240 (a number easily divisible by 2, 5, 10, etc.) and it encompasses the range of the actual data (13 - 248).

We must now choose a number of major divisions to use along the vertical or y-axis. 240 is evenly divisible by 24, but we do not have 24 major division along the vertical length of this paper. 240 is also divisible by 12. If we use 12 major division then each will correspond to 20 grams:

240g/12 major divisions = 20 grams/1 major division

and each minor division will correspond to 2 grams:

20g/5 minor divisions = 4 grams/1 minor division

Again we have numbers that are convenient to follow and allow for plotting the points to the precision of the data. Here we will have to estimate the nearest 1g/100g water as being exactly one fourth of the way between two of the minor.

Summarizing the process by which we arrived at this scaling of the axes:

1) We examined the range to be covered by each of the variables.

2) We chose a number of major divisions along each axis that was evenly divisible into the approximate range of the data.

3) By dividing the range by the number of major divisions we determined the value of the major divisions in terms of the units of measurements.

4) We determined the value of each minor division by dividing the value of the major division by the number of minor divisions between each major division, in this case 5.

Ordinarily, using integer values makes points on the graph easier to plot and read. At the same time we tried to utilize enough of the graph paper to make plotting the points as precise as the precision shown by the data. These steps were carried out before the axes were drawn in. You may have to try a couple of combinations before arriving at a satisfactory scale, but the effort is worth is when it comes time to plot the points. Also note that it is not necessary that the divisions along the x- and y-axes be the same value as long as each is clearly labeled as to the value of each division.

STEP 4: Assign values to the origin of the graph.

Table 2: Solubility of KNO3 as a Function of Temperature

Solubility of KNO3 (g/100 g water)

Temperature (° C)

13

0

21

10

32

20

46

30

64

40

86

50

110

60

138

70

170

80

208

90

248

100

Origin values:

10

0

An obvious possibility would be to make the origin (0,13). As a matter of fact, 0 for the x-axis is a reasonable choice, since this would make each major division along the x-axis correspond to a 10° increment in temperature. That is, the major divisions would read 0, 10, 20, 30… 100. On the other hand, 13 would not be a particularly convenient choice for the y-axis since then the major divisions would read 13, 33, 53, 73, etc. It would be easier if we made each major division an integer multiple of 10. There are a couple of ways to do this, but maybe the simplest would be to begin the y-axis at 10 (which is below the lowest solubility value). Then the major divisions would read 10, 30, 50, 70…250 (since here we determined that each major divisions would be worth 20g/100g water). The twelve major divisions will neatly encompass the data given in the table.

STEP 5: Number each major division along each axis.

Using the values determined for the origin in Step 4 and the values of the major divisions determined in Step 3, the scales on the x- and y-axes have been drawn in, as seen in the figure below. This should be an easy step if the two previous steps have been carried out with care. Also note that the actual data points do not necessarily appear as part of the numerical scales on the axes.

STEP 6: Complete labeling of both axes.

The labels for both axes have been placed on the graph, as seen in the figure below. This is an important step if the graph is to clearly communicate information. Both the quantity measured and the units used must be part of each axis label. Usually the units are placed in parentheses after or below the label for the quantity measured.

STEP 7: Title the graph.

Format: Figure (number): Brief description of graph

The graph is titled by assigning it a number. For example, Figure 1 is appropriate if it is the first graph to appear in a report. Thus the beginning of the title is the word "Figure" followed by an appropriate number. This is followed by a brief description of what the graph represents. At a minimum this should clearly specify both variables shown on the graph and somehow indicate which is the dependent and which is the independent variable. If any other conditions crucial to the experiment were adjusted or held constant these should also be specified in the title. In short, the title actually gives a brief description of the experiment represented on the graph. This means that some thought must go into writing the title for the graph

The brief description for the set of data we have been using might read "Solubility of KNO3 as a Function of Temperature." This adequately presents the variables and shows that the solubility was the dependent variable. Since there were no other factors which the experiment depended upon, this title is acceptable. (Note the title below for Figure 3).

STEP 8: Plot the data points.

Here, we will consider the actual plotting of only the first and second data points. In the set of solubility data, the first data point consists of a solubility value of 13 at a temperature value of 0. To find where this point belongs on the graph, we first locate 13 on the vertical axis. It will be between major divisions 10 and 30 and since each minor division is worth 2, it will lie between the first and second minor divisions above 10. Keeping this in mind, we try to locate the temperature value of 0 on the horizontal axis. As it happens we don't have to look far since we chose to have the temperature axis start at 0. A small dot is made to indicate this point, and a small circle is drawn around the point to indicate clearly its location. This is shown on the graph, as seen in the figure below.

To locate the second point (10, 21), we have to find where 21 falls on the vertical axis. This will again be between 10 and 30 and will be 5 and a 1/2 minor divisions above 10. To put it another way, this value will lie halfway between the divisions that represents 20 and the one that represents 22. Now, we move out from this point parallel to the horizontal axis until we come to the line corresponding to 10° . This is the first major division to the right of the origin and we plot the point in the same manner as was done for the first data point. Two data points are shown plotted on the graph. Continue in this manner until all data points are plotted.

STEP 9: Draw a smooth curve.

If we are lucky enough to have all the points fall on or near a straight line, drawing a smooth curve is an easy step. In such a case a straight edge (ruler), preferably one that is transparent (to enable one to see the points through the ruler), is used to draw a smooth continuous line that comes as close as possible to all the data points. Under no circumstances should a zigzag line be drawn in an attempt to force the curve to pass through every point.

Figure 4: Drawing a straight line through data points.

In the solubility graph that has been developed above, the plot is obviously not linear, at least in the lower portion of the curve.

With a nonlinear plot a French curve should be used to construct the smooth line. Here several examples of French curves are shown. These are usually made of transparent plastic and are placed on the graph such that some portion of the French curve connects two or three of the points

A smooth curve is then drawn along the plastic to connect the circles around the points.

Note that in either type of curve the line does not pass through the circles that pinpoint the data points but only touches the edges of these circles. This allows one to accurately locate the actual data points even after the line is drawn.

The curve is moved to coincide with the next couple of data points and the smooth line is carefully continued up along the data points. This process of moving the French Curve to coincide with several data points and drawing a segment of the graph is continued until a smooth unbroken curve ties all the data points together. Remember, the curve doesn't have to touch all data points; some points might be off the curve.

In using a French Curve you should avoid the common mistake of trying to draw too large a portion of the curve at one time. You will ordinarily have to shift the French Curve several times to avoid getting sharp breaks in the line. Notice on the graph that the line does not cut through the circles indicating the data points.

The results of the curve fitting process are shown for Figure 3 below.

Often you will notice with experimental data that all the points do not lie directly on the line but may be scattered above and below the line in a nearly equal manner. This scatter is due to experimental uncertainty and is illustrated by the third and fifth data points on the graph shown. The third point is somewhat below the line while the fifth lies slightly above the line. Generally, more precise measurements will yield graphs where more of the points lie directly on the line. This precision is usually indicated by the number of significant figures used to indicate the scaling of the axes. For example, the body temperature measurements on the graph in Figure 5 (below) were recorded to the nearest 1° . If these measurements had been made to the nearest 0.1° C, the numbers on the axis would have been written 10.0, 20.0, 30.0 etc.

With data of lower precision we frequently find that one or more of the points appear to be significantly off the curve. If there is good reason to believe that a particular point is in error, it is legitimate to ignore that point in constructing the line. (Whether or not you should extend the line or curve past the last point plotted depends upon the nature of the data.) When this step of the graphing is completed, you should have a graph that clearly, completely and accurately portrays the data. There should be no question in the mind of someone observing the graph what the graph represents.

Common Errors to Avoid

Often in the construction of a graph, one or several of the above points or steps are omitted or carried out incorrectly. Four examples of the more common errors that occur in the construction of graphs are reviewed in the next section. This may help you to avoid these common problem areas and at the same time reinforce or explain further some of the points already presented.

Example 1 (Figures 7a and 7b)

% Saturation of Hemoglobin

Oxygen Pressure (mm)

0

0

18

20

49

40

85

60

96

80

96

100

96

120

This curve (Figure 7a above) was constructed from the data shown in the table above. The data were taken in order to ascertain how the uptake of oxygen by blood hemoglobin is affected by the amount of oxygen (measured as a pressure of oxygen gas) to which the blood was exposed. However, if you examine the title of the graph and the units and labels on the axis, you would not necessarily know that this is what the graph is trying to show. In fact, from the title and the rather ambiguous labels all you would be sure of is that it shows percent saturation of something as a function of some pressure. Thus an important point to keep in mind is: The title and labels should be explicit enough that the graph can stand alone with little or no explanation of the experimental data.

Here the graph (Figure 7b above) has been labeled and titled so that these two portions of the graph convey the necessary amount of information.

Example 2 (Figures 8a and 8b)

Presented here is a set of data intended to be a curve that illustrates the growth of a current in an electrical circuit that contained inductance and resistance as a function of the time after the circuit switch was closed.

% of Final Current

Time After Closing Switch (msec)

0

0

35.0

1.2

63.0

2.4

88.0

4.8

96.0

7.2

99.0

9.6

Figure 8a above was constructed from these data.

A quick look at the graph seems to indicate that the current grows slowly at first and then increases sharply. However, a closer look at the labels on the axes shows that the dependent variable (% of final current) has been placed on the x-axis rather than on the y-axis, and that the independent variable, time, has been placed on the y-axis instead of the x-axis. Thus if you now look at the graph with this in mind, the current actually grows quickly rather than slowly in the initial stages. Complete labeling allows for the correct interpretation of the graph but without careful examination of these labels an incorrect conclusion about the relationship might be reached. Thus, it is important to make sure that correct identification of the dependent and the independent variables is made before the graph is constructed.

A second graph of the same set of data, shown in Figure 8b below, with the independent and dependent variables on the correct axes, more clearly shows the original intention of the graph.

Example 3 (Figures 9a and 9b)

Percent Salinity

Percent Chlorinity

5.2

2.76

10.1

5.56

14.8

9.38

20.1

11.21

25.0

14.08

30.1

16.95

34.9

19.86

The data in the table for the percent salinity and percent chlorinity of brackish and sea water have been placed on the graph, Figure 9a. It would appear from the data points that the two are related in a linear fashion, yet the line drawn on the graph is not a straight line but "dips" between the second and the fourth data points.

However, a line that fits the tendency of the data should actually be straight as shown BELOW in (Figure 9b). The line drawn should show the general trend of the data and does not have to pass through each individual point.

Example 4 (Figures 10a and 10b)

Remaining Mass of Ra (g)

Duration of Decay (days)

1.00

0

0.70

2

0.48

4

0.30

6

0.17

8

0.10

10

0.09

12

In this example, the data table and the graph (Figure 10a) represent the radioactive decay of the element radon with respect to time by measuring the amount of radioactive material left with the passage of time. The graph seems to indicate that this occurs in a linear manner, but if you examine the data table during the first two days. 0.30g have decayed (the remaining amount of radioactive material going from 1.00 g to 0.70 g) and during the second two-day interval (day 2 to day 4), 0.22g have decayed (the remaining amount of radioactive material going from 0.70g to 0.48g). If the relation is linear, then the change from one two-day interval to the next should be constant but the data shows that it is not. Therefore something is wrong with the graph. The error is the scale on the y-axis. Each of the major divisions along this axis should represent the same absolute amount, but if you look at the graph they don't. From the first number at the origin (0) to the next number (0.09) the major divisions appear to be worth 0.09g, but going from the second number (0.09) to the third number 0.10, the major divisions become worth 0.01 g, and so on up the axis. Thus the y-axis has an incorrect scale.

Using the format for scaling axes present earlier, the y-axis has been rescaled for the graph shown below in Figure 10b.

Note that it changes the nature of the curve and one's view of how radon decays. Scaling may be one of the more difficult steps in constructing a graph but from this example once can clearly see that correct scaling of the axes is indeed important.

When you feel that you understand the basic principles outlined for the construction of a graph, obtain a Post-test from the personnel in the Science Learning Center and the materials needed to construct the graph. Complete the exercise and turn it in to be checked. If your graph is correct make sure your name is recorded in the Science Learning Center database. If it is not correct, you may review this module and retake the Post-test as many times as needed in order to complete the exercise correctly.