The principles of graphing

Understanding the principles of graphs and their meaning are critical to understanding

economics.

The following material is designed to serve as a review on graphing. You have probably seen all this material before…many times. But it is important that you understand it as we

will rely extensively on graphs in this course.

If, after reviewing this material, you do not understand it, you are strongly encouraged to seek my help or consult the

tutoring center!

Slide 1 of 25

Where are we headed?

We’ll cover this material starting from the very beginning. Please bare with me.

Quickly, we’ll add details and complexity and demonstrate concepts

in an economic framework.

It is critical that you understand this material. You’ll see about 300 graphs

this semester and in each case, I’ll assume you understand it.

Slide 2 of 25

Let’s start with the number line

Here is a number line.

Also note that only whole numbers are shown. The number line is actually made up

of an infinite number of points each representing some fraction of the line.

Note how the arrows signify that numbers continue into infinity in both

directions.

Slide 3 of 25

½-3 ½

Two Variable Graphs (Cartesian Plane)

If we put two number lines together, then we get “space”. In this examples the number lines are referred to as the X axis and the Y axis.

Slide 4 of 25

Two Variable Graphs (Cartesian Plane)

(X,Y)

(3,2)

(4,0)

(-1,-2)

(-5,5)

Using those two axes, we can plot points on the “graph” using coordinates. Those coordinates are presented in an (X,Y) format.

Using your finger, predict where each of the coordinates below will be prior

to hitting enter.Y

X

Slide 5 of 25

Y

X

The upper right quadrantThe upper right quadrant

is the most commonly seen (in newspapers, etc)

because it contains positive values for both the

X and Y axes.

Note that the other 3 quadrants are still there…

they are just not shown because they are not of

interest.

Slide 6 of 25

7

Relationship Between Studying and GPA

Hours Studied Per Week Grade Point Average

16 4.0

12 3.0

8 2.0

4 1.0

0 0.0

Using graphs to determine the relationship between two variables

Using two axes allows us to compare two variables. As an example, let’s compare the variable “hours studied per week” with “Grade Point Average”

Notice how these look like XY coordinates? We can plot them (on the next slide) and see if a relationship exists between studying and GPA!

If you study 16 hours per

week, you might get a 4.0

If you study 4 hours per

week, you might get a 1.0

Relationship between hours spent studying per week and GPA

If hours studying per week were

displayed on the X-Axis and GPA were displayed on the Y-

Axis, the points would appear as

displayed.

If those points were connected, then we’d see an upward sloping line. This suggests a

POSITIVE RELATIONSHIP between studying and GPA (in other words, the more you study, the better your grades), which seems logical.

Here we have interpreted data on a graph. In this case, we have a direct or positive

relationships between variables. That is a key learning outcome!

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Different types of relationshipsExamine the

different types of linear

relationships.

The next few slides will show examples

of these.

Slide 9 of 25

Let’s practice! Try to identify these real-world linear relationship

What do you think the relationship between income and life expectancy is?

Positive, negative, or no relationship?

Generally speaking, it is positive. Note that each dot represents a country. In general, the higher the income, the higher the life expectancy. This can be seen by placing a line (in red) over the dots.

This may not seem fair, but it is true. People living in countries with higher incomes usually have better access to doctors and dentists and usually have access to a broader array of foods.

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What is the relationship between the unemployment rate and income?

Positive, negative, or no relationship?

Real-world application (part 2)

Generally speaking, it is negative. Note that each dot represents a county in the U.S. In general, the higher the unemployment rate, the lower the household income. This makes sense as employers in high

unemployment areas probably do not have to pay employees as much to attract them.

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Real-world application (part 3)

What is the relationship between education and income?

Positive, negative, or no relationship?

Fortunately, it is positive! Note that each dot represents a county in the U.S. In general, the higher the

education level, the greater the income. In other words, stay in school!!!

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Real-world (ridiculous) application (part 4)

What is the relationship between my height (6 feet) and population in the world’s countries?

Positive, negative, or no relationship?

Note that each dot represents a country in the world. As we would expect, there is no relationship between my height and any country’s population. Those two

variables are not related. Therefore, the red line that represents this relationship has no slope (it is flat)!

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Different types of relationships

Remember that there are 4 types of linear relationships!

Slide 14 of 25

Give it a shot!

• Determine two variables that could be measured. – Examples include income, hours worked, wealth,

education, population or many others. – Variables can also include non-economic

measures such as speed, distance, strength, time spent playing video games or many others.

• Then determine what you think the relationship between those two variables would be.– Appropriate answers will be either positive (also

called direct), negative (also called inverse) or no relationship.

• Continue this exercise until you have uncovered a positive and negative relationship and no relationship.

If you want, you can submit these to me via

email (slacroix@tcc.edu) and I’ll let you know if you

are on the right track!

Slide 15 of 25

Let’s talk about Non-Linear Relationships

Here are some examples of non-linear relationships

In some cases, relationships can be “non-linear”, meaning they aren’t shown as a straight line.

Slide 16 of 25

Real world example of a nonlinear relationship

Imagine we are comparing tax rates to tax revenue.

That means we are comparing the rate you charge people per dollar of income

to the total amount of money the tax generates.

I think you’d agree that in general, higher tax rates

result in higher tax revenue.

That would be true up to a point. At

some point, as rates get higher, we might

expect people to stop working or maybe they’d hide their

income.

This idea is referred to as the “Laffer Curve”. It is used to illustrate

the idea that the government can

maximize revenues by setting tax rates at some optimal point.

Think of it this way – if the government told you that the tax rate

was 100% - everything you earned must be

given to them… How much income would

you report!?!

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Let’s turn our attention to measuring the slope of a line

The slope of a line can be measured using this formula:

Or

Slide 18 of 25

Here is an example

In this case, a 4 unit increase in studying results in a 1 unit increase in GPA.

Therefore, the rise = 1 and the run =4.

So the slope of this line is ¼ or 0.25.

We know this relationship is positive because the slope is above zero!

This is the rise

This is the run

Slide 19 of 25

Example of calculating slope (part 2)

In this case, a 4 unit increase in beer consumption results in a 1 unit decrease in GPA.

Therefore, the rise = -1 and the run =4.

Therefore the slope of this line is -¼ or -0.25We know this relationship is negative because the slope is below zero!

Slide 20 of 25

Equation of linear relationship

y = a+bxWhere y= Dependent variable

a=vertical (y-axis) intercept

B=slope

x= Independent variable

Any linear relationship can be summarized using the following formula:

You may have learned this as “Y=mX+b”. If you understand that..that is fine. We use this formula because it is used in the textbook

Slide 21 of 25

22

Let’s take a look at that formula

Suppose we have the equation:

y=500+.5x

Then we can plug in values for x to get y

This (500) is “a” which is also called the “Y-

intercept”

This (+.5) is “b” which is

also called the “slope”

So if you pick a number at random (say $2000) and plug it into X, you

get

Y= 500+(.5*2000)…

which equals $1500

This equation may typify a person’s spending and can also be seen graphically

Note the positive slope indicating the positive relationship

between these variables. This

should make sense because the more

you make, the more your spend (typically).

In fact, we can now examine this relationship:

Since the Y-Intercept is $500 we know that

even if you do not earn any income, you’ll spend $500.

And since the slope is +.5, then for every dollar of income we earn,

we spend 50 cents!

y=500+.5x

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In summary

The relationship between any two variables can be illustrated graphically (which is a key learning outcome).

That relationship can be positive or negative, and linear or non-linear….or it

can have no relationship at all.

For linear relationships, we can measure the slope of the line using y=a+bx.

“a” is the y-intercept and “b” is the slope...they describe the relationship!

Slide 24 of 25

It’s negative. For each 1 unit increase in X, Y falls by 0.5!

Try this one!

Y=4 – 0.5X

Using this formula, graph the relationship

Start by selecting some (small) numbers for X and then solving

for Y. Click to see mine.

Then, once you have three “coordinates”, plot them below. Click to see the line.

Is this relationship positive or negative?

Slide 25 of 25