Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last several lectures we explored the concept of a "binary relation".
We saw that the set of ordered pairs which forms a binary relation
is sometimes referred to as the "graph" of the relation.
So in a binary relation on R, the graph would be a set of ordered pairs of real numbers.
Since Cartesian coordinates can be used to visually represent ordered pairs of real numbers
this is a way to visually represent the graph of a relation on R.
Graphing a set of ordered pairs can be useful
when trying to understand how two quantities may be related.
For example, could there be some correlation between the height and weight
of members of the U.S. Men's National Soccer Team?
Let's start by making a table with the height and weight of each player.
We can then create a collection of ordered pairs
where the first element of an ordered pair is the player's height
and the second element is his weight.
This collection of ordered pairs forms a binary relation.
The domain of this relation is the set of all height values
and the range is the set of all weights.
Notice that several players have a height of 72 inches
or a weight of 170 pounds.
But these values are listed only once in the domain and range.
This is because the domain and range are sets and every member of a set must be unique.
No member of a set may be listed more than once.
Now let's graph this relation using Cartesian coordinates.
For each ordered pair
we will plot the position of the first coordinate on the horizontal axis
and the second coordinate on the vertical axis.
Since height and weight can only have positive values
we can eliminate everything except the first quadrant of the graph.
We will mark the horizontal axis in increments from zero to 80 inches
and the vertical axis in increments from zero to 300 pounds.
Now starting with the first ordered pair
we locate the value of the first coordinate on the horizontal axis
and the second coordinate on the vertical axis.
So the point corresponding to the height and weight of the first soccer player
is located at this horizontal and vertical position.
Likewise, the ordered pair corresponding to the second player
is represented by a point at a horizontal position of 76 inches
and a vertical position of 210 pounds.
Each of the remaining ordered pairs is plotted in the same way.
Notice that these points are very close together
and take up only a very small portion of our graph.
However, we can rescale the graph so that the points are spread over a larger area
and easier to see.
When graphing only positive values it is not necessary to start the axes at zero.
For example, we could make the first division on the horizontal axis 60 inches
and the first division on the vertical axis 100 pounds.
Graphs of finite sets of ordered pairs are called "scatter plots".
Scatter plots allow us to visually detect patterns
which may exist in a relation between two quantities.
For example, in this scatter plot it is apparent that as a general trend
the weight of a player increases as the height increases.
We summarize this correlation between the two quantities
by drawing what is called a "trend line".
This example illustrates what in mathematics is called a "positive correlation".
since on the average, as a player's height increases, so does his weight.
But, if as height increased weight decreased
height and weight would be "negatively correlated".
Or, if no trend between height and weight was apparent
we would say they were "uncorrelated".
When the data points are grouped closely to the trend line
we say that they are "strongly correlated".
If the data points are grouped more loosely around the trend line 0:04:55.500,0:04:49.600 we say that they are "weakly correlated".
A trend line is sometimes called a "line of best fit"
and there are various ways to draw this line.
In general, a trend line is drawn in a way
that minimizes the distance between the line and the data points.
One way to draw a trend line
is to simply estimate the position which seems to fit the data points the best.
This is similar to trying to build a straight road
that comes as close as possible to several different towns.
The optimum path might be the one which minimizes the sum of the distances
between the road and the towns.
Many spreadsheet programs and scientific calculators
have functions that mathematically calculate trend lines
which minimize the vertical distance to the data points.
One of the most common methods is the "least squares" method
which minimizes the sum of the squares of the vertical distances.
Scatter plots can be useful for comparing characteristics of different groups.
For instance, if we graph the weight versus height
for the U.S. Men's National Soccer Team in blue
and the weight and height for a random sample of men in red
we would see that the weights and heights of the soccer team
were much more tightly grouped than the random sample.
One reason could be that the U.S National Soccer Team players
are in better shape than the average person.
One important type of binary relation which we will often encounter in mathematics
is a special type of relation called a "function".
In the next lecture we will see how to create functions
which will allow us to visualize and analyze many types of interesting and useful relationships.