Experiment 1:  Graphical Representation of Data

A line graph shows the relation between two variables in the form of a curve.  Graphs are often useful in determining the mathematical relationship between two variables.  Line graphs are used to compare values of two variables using two orthogonal straight-line scales called axes.  The vertical or y-axis is referred to as the ordinate, and the horizontal or x-axis is referred to as the abscissa.  If we have two variables x and y, and y is a function of x, expressed as  y = f(x), y is referred to as the dependent variable, and x as the independent variable. Values of y, the dependent variable, are plotted on the ordinate. Values of x, the independent variable, are plotted on the abscissa.  When plotting experimental data one should be careful in choosing the appropriate variable and thus the appropriate axis.  In general, the independent variable is altered in regular intervals in the experiment, and the dependent variable is the quantity measured or calculated for each regular step of the independent variable.  When drawing a graph, observe the following rules:

1.      Mark the scales along the axes and label each scale with the variable being plotted.  Units should be included in parentheses.  The graph should be easy to read. Make each large division equal to an appropriate scale to facilitate plotting a readable graph.  Plotted data should fill most of the graph, and should not be confined to small area of the graph. Where appropriate, error bars should be included. Last, the point (0,0) need not appear on the graph as a point unless it is a relevant point on the graph.

2.      Plot points carefully.  Use suitable symbols for each point. When two different sets of data are plotted on the same graph, use different data symbols for each data set. Indicate in a legend what experimental conditions correspond to each data set.

3.      If appropriate, fit a curve to the plotted points.  Most graphs will be for the purpose of verifying a law or determining a functional relationship.  Graphs will show either a uniformly smooth curve or a straight line.  Many graphing programs will automatically draw a curve for your data set.  Include on the graph the equation of the best fitting curve and the R2 value. The closer the R2 value is to unity (one) the better the data set.

4.      Give the completed graph a title.  The title should take the form of “Y versus X under the conditions Z”.

# Interpretation of Data

Many graphs you encounter will have variables chosen so that the graph itself is a straight line, and the linear relationship:

y= m x + b

where m is the slope of the line and b is the y-intercept.  In graphs drawn from experimental data, the constants m and b have physical significance and may have specific units associated with them.   Note that plots of this type have value in that they may be used to predict data values given certain input data.

Sometimes graphs may be used to explore a trend, or discover a relationship between two variables.  The plots may not necessarily be linear in nature.  It is up to the investigator to decide what type of function best fits the data points.  This can be done by trial and error. A function that fits the data well will have an R2 value close to unity.  In  exercises 3 through 5 you will explore data trends for the periodic table of the elements.

Exercise 1.

The following relationship allows you to convert from the Fahrenheit scale to the Celcius scale:

OC = 5/9 oF – 5/9(32)   note that it takes the form y= mx +b

A student records the following data. Determine the ordinate and abscissa, dependent and independent variables.   Plot the following data set and cite appropriate units.   Include a title, and show the R2 value and formula on the plot.  How well does this data set correspond to the formula? Compare the intercept and slope to the theoretical equation above.   Do the equations match exactly?  What limits are there in precision?  How well does the trend line fit the data?  Explain in detail.

## Celcius

-40

-40

-20

-28.9

-15

-26.0

-5

-20.9

0

-17.9

5

-15.0

20

-6.9

80

26.9

200

93.5

400

204.5

Exercise 2.

Plot  the following data for thermal expansion of a sample of oxygen gas at 1 atmosphere constant pressure.  Plot the data citing units and using appropriate labels. Determine the slope and intercept and write the equation for this line plot.

 Volume  in Liters Temperature in Celcius Degrees 25.00 31.50 30.00 92.39 35.00 153.30 40.00 214.17 45.00 275.20 50.00 336.11 65.00 ? Predict this value using your plot

Refer to the periodic table for the following exercises.

Exercise 3.

Plot the atomic mass vs. the atomic number for all elements in column 1 of the periodic table.  Assume that the plot passes through (0,0).  Cite appropriate units and   include a title.   Show the R2 value and formula on the plot.

Exercise 4.

Plot the atomic mass vs. the atomic number for all elements in period 4 of the periodic table. Assume that the plot passes through (0,0).  Cite appropriate units and include a title.   Show the R2 value and formula on the plot.

Exercise 5.

Plot the density vs. the atomic number for all elements in period 4 of the periodic table.   Cite appropriate units and include a title.   Show the R2 value and formula on the plot.  Do a similar plot for all elements in period 5 of the table and compare your results.

## Questions:

1. Is there a general trend in the molar mass vs. atomic number plot? The slope of the plot is not equal to 1. Why does the molar mass increase by more than one unit when the atomic number increases by one unit?

1. Is there a general trend in the density vs. atomic number plot? Does the pattern of the plot repeat? What does the term periodicity mean to you? Explain in detail.

Graphing with Excel:

1. Open Excel – a worksheet should open.
2. Type in your data points – X axis is usually in column A and Y-axis is usually in column B.
3. Click on the chart wizard button on the toolbar.  (It looks like a bar graph.)
4. To plot you points as on single line, click on the XY Scatter choice for chart type.
5. Under chart sub-type, choose either the curved line or straight line choices.  Click on the next button.
6. Under data range and series, leave at the default values and click next.
7. Under the titles tab, add the chart title and titles for the X and Y axis.
8. Under the axes tab, make sure that under primary axis both value x axis and value y axis are checked.
9. Under the gridlines tab, choose the configuration of gridlines that you are looking for.
10. Under the legends tab, choose the placement of your legend information.
11. Under the data labels tab, choose show label.  This will give each data point value for the x-axis.  If you choose show value, it would show the x-axis value for each data point. Click next.
12. Under place chart, choose as a new sheet – chart 1.  This will save the graph as a full page and not a smaller version of the graph.  If you choose “as object in – sheet 1”, the graph will show up as a smaller graph right on your data sheet.  Click finish.
13. If you right click on any of the titles (chart title, axis titles, legend), you can modify them to suit how you want your graph to look.
14. If you right click on the x-axis numbers, you can change the scale or modify the information shown on the x-axis.
15. If you right click on the y-axis numbers, you can change or modify the y-axis information.