Topic 2: Bivariate Data

Two quantitative variables have a positive association if larger values of one variable tend to occur with larger values of the other variable. The variables have a negative association if larger values of one variable ten to occur with smaller values of the other variable.

Does the Super Bowl data show any positive or negative association (it doesn't have to show any...)?

In mathematics, a general pairing of quantitative variables is called a relation or relationship. In ordinary language, when we say one entity is a function of another, we generally mean that the first is related, and in some way dependent on the second. In mathematics, the term function has a very precise meaning, it is a relation in which each value of the explanatory variable is paired with exactly one value of the response variable. In a function, we typically use x for the explanatory variable and y for the reponse variable.

Functions can be represented in many ways:

• tables
  

  Year (x)	Cases of Tuberculosis (y)
  1953	53.0
  1960	30.8
  1970	18.3
  1980	12.3
  1990	10.3
  2000	5.8

  
  
• symbols
  y = f(x)
  
  
• graphs
  
  


• words
  "A 150 pound person uses approximately 5.4 calories per minute walking at a normal pace on an asphalt road."

In some functions, the response variable is equal to a constant multiplied by the explanatory variable. If this is the case, the constant is called the constant of proportionality and the response variable is said to be directly proportional to the explanatory variable.

Examples of proportionality:

• "The ratio of the radius of a circle to its circumference is 2*Pi"
  
• y = 4*x
  
• f(x) = (1/4) * x