Guidelines for Graphing

Often the goal of an experiment is to find the relationship between two variables (pressure and volume, time and temperature, etc.). As one variable changes, so does the other. Graphing is a useful way to visualize and describe these relationships. Because the use of graphs is so common in the sciences, it is important that you know how to construct and interpret graphs. When preparing graphs as part of chemistry lab reports at the University of Oregon, keep the following guidelines in mind:

  1. Computer generated graphs are generally preferable. If drawn by hand, graphs must be neatly drawn, using a straight edge, French curve, or flexible curve.

  2. Tabulate all data to be graphed before beginning.

  3. Use the x-axis for the independent variable (that which is experimentally varied; also known as the manipulated variable) and the y-axis for the dependent variable (that which is a function of the independent variable; also known as the responding variable).

  4. Decide on the limits of the graph (maximum and minimum values) and select the axis. The lower left corner of the graph does not have to represent zero on either axis unless you have data in this region.

  5. Select divisions on the axes which are easy to read. Common graph paper is divided in units of tens. Hence one square may equal 1, 2, 5, 10, or 10, 20, 50, 100, but never 3.75 or some other "odd" number.

  6. For greatest accuracy, select scales so that the graph nearly fills the page.

  7. Grid lines should be shown on the graph. (If drawn by hand, all graphs are to be drawn on graph paper with at least 10 squares per inch. The gridded pages (1/4" quadrille) in your lab book are not adequate.)

  8. Label both axes with both quantity and units. For example: "Pressure (torr)"

  9. Mark the data points with a small dot. (Pencil may be used before being inked for permanency.) Draw a small circle around the point or darken to make more visible.

  10. A smooth curve should be drawn through the points. The curve should pass as close as possible to each of the points but should not be connected point-to-point. (The data may be irregular but nature rarely is.) If the relationship appears to be linear, the smooth curve should be a straight line. If the line is extended past the range of the measured values, this extension should be indicated by a dashed rather than a solid line.

  11. Title the graph in a descriptive manner.

Linear Relationships

Recognizing a pattern in data is helpful but generally not enough. It is even more useful to develop a mathematical equation that fits the data. This then allows us to calculate the value of the dependent variable at any value of the independent variable.

If the plot of the data gives a straight line, we can say that the dependent variable (plotted on the y-axis) is directly proportional to the independent variable (plotted on the x-axis) In that case, we can then fit the data to the equation for a straight line,

y = mx + b
where m is the slope of the line and b is the y-intercept.

In some instances, a straight line arises when the data are fitted to the equation

y = m * (1/x) + b
The variables are then said to be inversely proportional. It can be seen that when there is a linear relationship between the two variables, the slope is the constant factor which relates the variables, where
slope = (delta y)/(delta x)
(Note: to minimize error when finding the slope, choose two points on the line which are not original data points.) The y-intercept is simply the value of the dependent variable when the independent variable is zero. Be aware that this may not necessarily be the point where the curve intercepts the y axis on your particular graph. Once these two values have been determined, the equation for a straight line can be used to find the value of y for any value of x.

The values of the slope and y-intercept are generally difficult to obtain directly from the graph. This is due in part to the difficulty in "eyeballing" the best straight line through the points, and also because the slope and intercept should represent the same level of precision as the data. (It is frequently impossible to assign the appropriate number of significant figures to a bet fit line which has been drawn by hand). A better approach is to apply statistics to define the most probable straight line fit of the data. For all linear plots prepared for the chemistry laboratory courses, you will be expected to determine the slope and y-intercept using the method of linear regression analysis (or method of least squares).

A linear regression analysis determines the best fit straight line through the points by minimizing the sum of the squares of the deviations of the points from the line. This analysis of the data can be done quickly using a computer spreadsheet program or graphing calculator. (Instructions are available on the course web page for performing a linear regression analysis using TI-81, TI-82, and TI-85 calculators.) Remember that your eyes are smarter than your calculator and you must always include a graph in your lab report. This graphical representation of the data allows you to visualize the relationship between the variables, see the "scatter" in the data, and consider whether "bad" or questionable data exists.

Reading the Graph

Remember that the precision of the information obtained from the graph should match the precision of the data that went into the graph. It is not desirable to lose significant digits when reading the graph, nor is it possible to generate more. Therefore, if the data used to generate the graph had N significant figures, numbers read from the graph should also have N significant figures. This rule holds true as well for values of the slope and y-intercept obtained by linear regression analysis. It will generally be necessary to round the values obtained from your calculator or computer program to the correct number of significant figures.