Finding Patterns in Measurements

## Line Graphs ### Background Information

For our distant ancestors, recognizing whether they were about to step on grass, off a cliff, on slippery mud, or into a pond probably was critical information.  Perhaps as a result, the human brain has evolved an ability to recognize visual patterns.  Since we have this ability, it provides us a valuable tool for analyzing other kinds of information.

Recall that Pythagoras proposed that mathematics and numbers revealed the essence of the universe.  Plato taught that mathematics could be used to explain the universe.  Their followers were increasingly successful in measuring and understanding aspects of the physical world then learning to use that understanding to control the world to their benefit. As their society emerged from the Midevil period, some Europeans such as Thomas Bradwardine (c. 1295-1349) tried to clarify the causes of change such as motion.  He and other fellows at Merton College searched for geometric methods of visualizing mathematical patterns for speed and other properties that change.  They began to draw bars and histograms of lengths proportional to successive measurements.  Nicole Oresme of Paris used what today would be called graphs to quantify physical qualities such as speed, displacement, temperature, whiteness, and heaviness, but also nonphysical qualities such as love, charity and grace.  For example, he used the geometry of a graph to prove a uniformly accelerated object travels the same distance as it would, had it travelled steadily at the average (Sav) of its initial (Si) and final (Sf) speeds.  (The area of the distance rectangle is equal to the area of the trapezoid because moving the pink triangle doesn't change its area.)

That search for patterns in the phenomena of our world continues with efforts such as Stephen Wolfram's A New Kind of Science in which he uses computers and a program called Mathematica to seek patterns in the apparent chaos of vast collections of related numbers.

While histograms, pie charts, and other bar graphs are helpful in visualizing patterns in one aspect of nature, we are often more interested in how one aspect of nature relates to another aspect.  For example, a fireman needs to know how the height water can be delivered from a fire hose depends on the water pressure.  Such information can be visually presented by a line graph.  Traditionally such a graph is made manually with pencil and graph paper.  But computer programs using spreadsheets such as Excel can also generate such graphs.  Learning to manually draw good graphs with all the necessary parts is a valuable precursor to manipulating Excel to make similar graphs.  Incidentally, making a pretty graph using a computer program such as Excel often requires more time than to make a good graph manually!

At this point, many people believe they have previously learned how to make graphs, skip further instruction, then construct FLAWED graphs!  A word of advice: Don't assume you already know everything; at least read the links below about making graphs and search for construction details that make a pretty but useless graph into effective communication.

#### How to Manually Construct a Line Graph

2. Identify variables
1. Independent Variable (controlled by he experimenter):  Values for this property are assigned to the horizontal axis ( ↔ often called the x axis)
2. Dependent Variable (This property should slavishly change depending on the value chosen for the independent variable.)  Values for this property are assigned to the vertical axis ( ↑ ↓ often called the y axis)
3. Determine the range of the two variables.  For each variable subtract the lowest data value from the highest data value.  If you wish the origin of the graph to start at zero, use the highest data value for the range.
4. Determine the scale of the graph.  Count the number of squares horizontally and vertically on the graph paper.  Divide the range of the independent variable by the horizontal number of squares, then round off to the next larger convenient number.  (For example, if dividing gives 34.3, it might be convenient to allow each horizontal square to have a value of 50.)  Determine the scale for the dependent variable by dividing its range by the number of vertical squares, then rounding up.  This procedure should spread the graph to use MOST of the available space.
5. Number each axis.  Starting with a number just smaller than the smallest independent data value (or zero if that is desired for your origin) label each LINE along the bottom of the graph from left to right by successive multiples of the chosen scale.  (For our example, 0, 50, 100, 150...)  It is critical that these numbers represent lines and not boxes.  That will allow the space between the lines to represent values in between.  Number upwards the lines along the left side of the vertical axis using the scale chosen for the dependent variable, starting just below the smallest data (or zero if so chosen for the origin).  If lines are so close together that the numbers will be crowded, omit writing some of the numbers but continue to space as if all lines were numbered. 6. Label each axis.
1. Briefly describe the properties represented by the independent and dependent variables.
2. In parentheses abbreviate the measuring units used to measure the data along each axis.
7. Plot the data points.  For each pair of related data, use the value for the independent variable to plot horizontal location and the value for the dependent variable to plot vertical location using the number lines along the axes.  Make a tiny dot at the intersection of the horizontal and vertical locations to accurately show the two values.  To make the approximate location of the dot apparent later, place a marker such as a small circle around the dot.  If other sets of data are also plotted on the same graph for comparison, use a different marker (square, triangle etc.) for each set of data and use a legend box to show what each marker represents.
8. Draw the line.  Most graphs of experimental data are NOT drawn by connect the dots.  Recall that all measurements contain experimental error, so as a result, most dots are not exactly where they should ideally be located.  To compensate for this experimentally error, try to draw a straight line that compromises, coming closest to nearly all the dots.  Sometimes a straight line does not fit well.  In that case, consider
1. drawing several intersecting straight line segments, each of which fits much data.
2. drawing a smooth curve that comes closest to nearly all the data.
9. Add a title, completion date, and author's signature.  The title should be selected to clearly but briefly tell what the graph is about.  Avoid cute teasers that attract attention but leave out critical information.

Construct a line graph for data such as provided in Table #1 below.

 Time (minutes) Distance (miles) Time (minutes) Distance (miles) 0 0 12 6.7 1 0.2 13 7.6 2 0.6 14 8.5 3 1.1 15 9.3 4 1.6 16 10.3 5 2.2 17 11.3 6 2.6 18 12.1 7 3.0 19 13.0 8 3.6 20 13.9 9 4.2 21 14.7 10 4.9 22 15.5 11 5.8 23 16.4

#### How to Construct a Graph Using a Computer

Nearly all computers have applications for constructing spreadsheets.  Such applications usually are capable of constructing graphs from series of data entered into the spreadsheet.  If you are working on a computer that uses an operating system from Microsoft, you probably have a spreadsheet program called Excel.  Jim Askew at Howe High School in Howe, Oklahoma has created instructions for making graphs using Excel.  To learn more about making effective line graphs using Excel, use this link to Jim Askew's instructions.

Using a computer spreadsheet such as Excel, construct a line graph for the data in Table #1 above.

### References

• J.A. Weisheipl, The Development of Physical Theory in the Middle Ages, Sheed and Ward, NY, 1959.
• I.B. Cohen, The Birth of a New Physics, Doubleday, NY, 1960.
• RCBrill's How to Construct Line Graphs, Honolulu Community College
• Lowell Boone's additional tips for Making Graphs, Physics Dept., U. of Evansville to Physical Science experiment 3-2