Line.html

Of all the possible graphs, far and away the most popular among economists is the line graph. Unfortunately, it is not one of the easier graphs to understand, ranking well behind the bar graph and pie graph on an ease of interpretation scale, but substantially ahead of the scatter diagram. Given that it is an important and frequently used tool and the lack of a widespread understanding of the tool, we'll spend a little time working on the basics of the line-graph.

The first step in creating a line graph involves specifying the axes of the graph. There are two number lines that are perpendicular to each other (blue lines). Each point on the graph represents one combination of values for the two variables. In the graph below you will see two points. If we assume that X is being measured on the horizontal axis and Y on the vertical axis, then at point #1 the value of X = -3 and the value of Y = 5. At point #2, X = 4 and Y = -2.

Now let's answer the question: What happens to the values of X and Y as we move from point #1 to point #2. First you should note that the direction is down and to the right. Down means less of Y while to the right means more of X. The movement from #1 to #2 therefor is a visual representation of an increase in X and a decrease in Y. Since they move in opposite directions, you would say that they were negatively related.

Now let's talk about some graphs. The place to start is by recognizing that behind every line graph is a table and behind every table is a story. So let's start with a simple story and work our way to a table and then to a graph. The story is about grades and the relationship between grades and study time. At this time in your educational career you probably believe that there is a positive relationship between grades and time spent studying. To make life easy, let's make it a very simple relationship. If you study no hours, you get a zero and for each hour studying you get a 5 point increase in your grade. What would the table and the graph look like?

The table and the graph appear below. In the table we can easily check out the story. As we move down one row we are looking to see what happens with one more hour of study and we see that the grade increases by 5 points. We get the same number whenever we compare two rows - it is always 5 more points for another hour of study? What we have is a constant rate of increase - every time we increase time spent studying by one hour, the grade increases just as fast.

Time	Grade
0	0
2	10
4	20
6	30
8	40
10	50
12	60
14	70
16	80
18	90
20	100

So what does the picture look like? Don't cheat and look ahead. First see if you have an image of what the picture of the relationship would look like. You know that this is a positive relationship - when study time increases the grade increases. What will you see in the graph that conveys this information? Now its time to look.

What you see is a line that slopes upward from left to right. This is the "picture" of a positive relationship. Each point on the line corresponds to a line on the table. The red line corresponds to the row with ten hours of study time. If you read over from the graph you see that at 10 hours of study time, the grade will be 50.

Now let's look at the slope - how steep the line is. A steeper line is said to have a greater slope, but what does that mean in terms of our story? Let's look at the graph below and work back to a table and then a story. We see that the line is still positively sloped so we know that there is still a positive relationship. We also see that the curve is steeper which tells us that the improvement in the grade for each hour spent studying is now greater.

How do we know that? From high school algebra you know that slope = rise/run. So let's pick two points. The first is the origin where the grade is zero for zero hours spent studying. The second is at five hours of studying. In the old graph (Blue line), the grades is 25. The grade on the new graph (green line) is 30. The slopes of the two lines would be:

The steeper slope means that each hour spent studying in the new situation increases the grade by 6 points. We would draw a steeper slope if we want to demonstrate a situation where changes in X (hours) created greater changes in Y (grades).

You have now worked both ways - translating from words to graphs and from graphs to words. This is a skill that comes with practice and you should consider getting some practice if you are not real comfortable with the translations.

For some practice consider the following two "extensions" of the model. The first is a nonlinear situation - a line that is not straight. To see what we have here let's try to create a table and a graph that correspond to the following "story." You will get 14 points without any study time and the first two hours spent studying will give you 13 additional points, the second 2 hours will give you 12 additional points, the third two hours will give you an additional 12 points, ... You can see the pattern. What will the table and the graph look like?

Time	Grade
0	14
2	27
4	39
6	50
8	60
10	69
12	77
14	84
16	90
18	95
20	99

Let's look at the table first and see what the table tells us about the rate of change. When we increase study time from 0 to 2 hours, the grade increases by 13 points. When you increase from 10 hours to 12 hours the grade increase by 8 points. What has happened to the rate of increase? The rate of change, or slope, for the two points are:

The rate of increase has gone down. In fact as we increase the hours spent studying, the improvement to the grade continues to fall. The rate at which the grades increase declines as the time spent studying increases.

Now how do we see this on the graph? Let's look. The change in the rate of change will show up as a change in the slope. Because the rate of change slows as we increase hours studying, the slope decreases as we move to the right.

Now to that second extension. When we draw a graph showing a relationship between two variables (Grade and hours), this does not suggest that there are not other factors that influence the relationship. Can you think of other factors that might alter the relationship between hours spent studying and grades? How about amount of serious drinking? I suspect that heavy drinking would decrease the value of your time studying. What about watching TV while you are studying? How about a study group? And what about a scale? What about the type of questions.

We'll look at the effect of the type of questions. Let's assume that you will get a mix of questions with more easy questions. What will happen to the relationship between the grades and the time spent studying? You should get a higher grade with the new questions - precisely what you see in the table below. Regardless of how many hours are spent studying, the grade is higher for the easy test. So how does this look on the graph?

Time	Difficult	Easy
0	0	5
2	10	15
4	20	25
6	30	35
8	40	45
10	50	55
12	60	65
14	70	75
16	80	85
18	90	95

The entire curve shifts. When one of the "other" factors that influences grades change, the relationship between grades and hours changes which shows up as a new line.

So much for the basics. You can now get some practice in the next two unit where we will be looking at possibility curves and supply & demand graphs. To convince your self that you have the basics of the graphs, you should make a serious effort to answer the questions of the day for the data analysis section. You should also check out the section on the scatter diagram.