Review
• association: an association exists between two variables if a
particular value for one variable is more likely to occur with
certain values of the other variable.
• response variable & explanatory variable
• positive association & negative association
• (categorical variable, categorical variable) — contigency
table
• (quantitative variable, quantitative variable) — scatterplot
• correlation
the correlation, r , summarizes the direction of the association
between two quantitative variables and the strength of its
straight-line trend. Denoted by r , it takes values between −1
and +1.
y − ȳ
1 X
1 X x − x̄
zx zy =
r=
n−1
n−1
sx
sy
where n is the number of points, x̄ and ȳ are means, and sx
and sy are standard deviations for x and y . The sum is taken
over all n observations.
• quadrant: a quadrant is any of the four regions into which a
plane is divided by a horizontal line and a vertical line.
Properties of the Correlation:
• The correlation r always falls between −1 and +1.
• The closer the absolute value of r to 1, the stronger the linear
(straight-line) association, as the data points fall nearer to
a straight line.
• A positive correlation indicates a positive association, and
a negative correlation indicates a negative association.
• The value of the correlation does not depend on the
variables’ units.
• Two variables have the same correlation no matter which
is treated as the response variable.
• The correlation is designed for linear associations
(straight-line relationships).
• Regression Line: an equation for predicting the response
outcome.
The regression line predicts the value for the response
variable y as a straight-line function of the value x of the
explanatory variable.
Let ŷ denote the predicted value of y . The equation for the
regression line has the form
ŷ = a + bx
In the above formula, a denotes the y-intercepr and b
denotes the slope.
• The y-intercept is the predicted value of y when x = 0.
• The slope b is the amount that ŷ changes when x changes by
one unit.