# Graphing

## Sketching Graphs by hand:

You may be required to graph by hand.

Sketching a graph

Sketched graphs are meant to just show trends, so you do not put data points, units or scale values. You should still have axes labels, possibly a zero y axis shown for reference, and a trendline.

See the examples of sketched graphs (as opposed to complete graphs below):

## Complete graphs drawn by hand

### Complete graphs should have the following elements, whereas sketches of graphs can be simpler.

• Axes titles (with units)
• Most often we graph dependent variables on the y axis and independent on x, but on occasion we find it useful to plot variations on this.
• Axes scales (numbers that tell how much each box is worth)
• Your data points should span at least 1/2 of the graph space on each axis. If not, you should consider changing your scale. I have a method for determining scale below.
• PLEASE do not just label your X values from your data set along the X axis. This will always make a linear graph, but it will always be meaningless.
• Data points
• each data point should be plotted as a dot. Be very careful in making sure that your read your scale correctly.
• If applicable, scale should be chosen so you should be able see x and y intercepts
• If linear draw a best fit line
• If there are asymptotes, they should be identified

How to scale a graph by hand:

1. Find the spread you need to show on your graph.
• I often use zero and the maximum value. In the data set at shown by the good graph example I chose a spread of 27.5 on the Y axis and 7 on the X.
2. Count the number of vertical and horizontal spaces on your graph.
• On the grid used to graph in the example I counted 15 spaces each way.
3. Divide the spread by the number of spaces on each axis and then round up to a convenient number to work with.
• For the Y axis 27.5/15=1.8, I rounded up to 2 and made each box worth 2 s. I chose not to label every line because that would be too crowded.

Be careful: Do not take uneven intervals (like 0.15, 0.40, 0.65) and plot them evenly on an axis. (See y axis on bad example below)

## Using Google sheets to create graphs:

Before starting to graph you should have your data prepared - - this video shows how to calculate values needed and format them to make it easy to graph.

These steps will help you create a scatter plot with trend lines, labels, equations, and an R^2 value to compare precision.

Use the Insert menu and select Chart to start creating a graph. Follow the following steps to format the graph:

### 1. Select your data

Under the Data tab of the chart editor select the data by doing the following:

• set the chart type to scatter chart
• set the series to the Y axis values (click add series and then highlight data with header; remove any excess series shown)
• set the X-axis to your X axis values (click Add X-axis and then highlight data with your header)
• the only box needed to be checked below is Use column A as labels

### 2. Add labels

3. Under the Customize tab of the chart editor expand the chart & axis titles settings to add axis labels:

• Add a chart title if desired
• Change the type to Horizontal axis title and enter the measurement and unit for your x-axis values (example: time (s)) as the horizontal axis title.
• Repeat with Vertical axis title

### 3. Add trendline

4. Still on the customize tab, expand the series settings.

• Check the box for the trendline
• Change the type between linear, polynomial, and power to find the best fit.
• Change the label to Use Equation to show the equation on the chart.
• Check the box for Show R^2 to display the how well the line matches your data (closer to 1.000 is good, 0.995 and up indicates that the line is a good model for the data set).

### 4. Check scales

4. Still on the Customize tab, expand the Horizontal Axis settings

• Make sure the box Treat labels as text is not selected. This will cause a non-linear scale
• If the default zoom over-emphasizes differences or fails to show an intercept, adjust min or max values to represent your data better.
• Repeat with the Vertical axis if needed.

### 5. Double check all required elements

Graphs need

1. Labels with units for each axis
2. Consistent scales that allow all data/analysis points to be seen
3. Trendlines (if data displays a trend)
4. Equation of trendline (if trendline is appropriate)
5. R^2 value (if trendline is appropriate)

## Interpreting Graphs:

• You should be able to describe graphs relationships by looking at them and matching them to the above trends
• A dependent relationship just means that one changes the other. This can describe all relationships except constant (not shown above, but where the y value does not change the X or vice versa).
• A positive correlation indicates that when one value increases the other does as well.
• A linear relationship is represented by a linear function y=mx+b
• A direct also known as proportional relationship is a special kind of linear relationship in that the b value is zero. so, y=mx. It must pass through the origin.
• A negative correlation indicates that when one value increases the other decreases.
• An inverse relationship (also called inversely proportional) follows the equation y=m/x or y=m(1/x).
• There is debate about what indirect means and I would avoid it's use. Some people consider it the same as inverse while others use it for any negative correlation.
• An inverse square relationship is tough to tell from a simple inverse relationship - you need to linearize it to tell the difference.