AP Statistics
Chapters 3 & 4
Measuring Relationships Between
2 Variables
Basic Terms




Response Variable: Measures an outcome of a
study.
Explanatory Variable: Helps explain or influences
changes in a response variable.
Scatterplot: Shows the relationship between two
quantitative variables measured on the same
individuals (one variable on each axis).
We are examining relationships and
associations. DO NOT ASSUME that the
explanatory variable causes a change in the
response variable.
Interpreting a Scatterplot


Just like with univariate data, we are looking
for an overall pattern and for deviation from
that pattern.
Overall pattern
– Direction: Negative or positive association
– Form: curved or linear? Are there clusters?
– Strength: How closely do the points follow a
clear form?

Deviations
– Outliers: Individual value that falls outside of the
overall pattern.
Correlation

Correlation (r) measures the direction and
strength of the linear relationship between
two quantitative variables.
– Does not distinguish between explanatory and
response variables (i.e. r would stay the same if
you switched the x and y axes)
– r has no units of measurement (correlation will
not change if you change the units for either of
the two variables)
Correlation

(+) r indicates a positive association
– As one variable increases, so does the other.

(-) r indicates a negative association
– As one variable increases, the other decreases.

r is always between -1 and 1.
– If r is close to zero, then the linear relationship is
weak.
– If r is close to 1 or -1, then the linear
relationship is strong.
Correlation
 CORRELATION
DOES NOT
IMPLY CAUSATION!!!
𝑟
𝑟
=
1
𝑛 −1
𝑧𝑥 𝑧𝑦
=
1
𝑛 −1
𝑥𝑖 − 𝑥
𝑠𝑥
𝑦𝑖 − 𝑦
𝑠𝑦
Regression line

A line that describes how a response
variable, y, changes as an explanatory
variable, x, changes. It is often used
to predict y given x.
– y = a + bx
b slope: the amount by which y changes on
average when x changes one unit.
 a y-intercept

Making predictions with
the regression line

Interpolation
– Estimating predicted values between
known values.


(Good )
Extrapolation
– Predicting values outside the range of
values used to make the regression line.

(Bad )
Least-Squares Regression
Line

Line that makes the sum of the
squared vertical distances between the
data points and the line as small as
possible.
– ŷ = a + bx
(ŷ  y-hat)
Slope: b = r(sy/sx)
 Passes through the point (x, y)

Example

An SRS of 50 families has provided the
following statistics
– # of children in the family

Mean: 2.1, std dev: 1.4
– Annual Gross Income

Mean: $34,250, std dev: $10,540
– r = .75

Write the equation for the least squares
regression line that can be used to predict
gross income based on # of children.
– Be sure to define your variables.
Residuals

Residual: The difference between an
observed value of the response variable and
the value predicted by the regression line.
– Residual = observed y – predicted y
=y–ŷ
Standard deviation of the residuals:
s=
residuals2
n-2
How well does the line fit
the data?


To answer this question, you must
look at two things.
1. Residual plot: scatterplot of the
regression residuals plotted against
(usually) the explanatory variable.
– If the regression line represents the
pattern of data well, then…
The residual plot will show no pattern.
 The residuals will be relatively small.

How well does the line fit
the data?

2. Coefficient of Determination: r2
– The fraction (%) of the variation in the
values of y that is explained by the least
squares regression line of y on x.

Template:
–
r2
% of the variation in (y-variable)
is explained by the least squares
regression line with (x-variable) .
Other Considerations
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
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Outlier: Observation that lies outside the
overall pattern (may or may not have a
large residual).
Influential Point: Observation which, if
removed, would greatly change the
statistical calculation.
Lurking variable: An additional variable that
may influence the relationship between the
explanatory and response variables.
Correlation v. Causation

The goal of a study or experiment is often
to establish causation…a direct cause and
effect link.
– Lurking variables make establishing causation
difficult.

Common response: Observed association
between two variables, x and y, is explained
by a lurking variable, z. Both x and y
change in response to changes in z.
Correlation v. Causation

Confounding: Occurs when the
effects of two or more variables on a
response variable cannot be
distinguished from each other, (often
occurs in an observational study).
Establishing Causation
w/o an Experiment




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1. The association is strong.
2. The association is consistent.
3. Larger values of the explanatory
variable are associated with stronger
responses.
4. The alleged cause precedes the
effect in time.
5. The alleged cause is plausible.
Non-linear relationships


If data follows a non-linear form, we
can sometimes transform the data to
become linear. By doing so we can
then perform the same analyses that
we do for linear data. (regression line,
correlation, r2, residual plot).
What are the most common non-linear
models for bivariate data?
Transforming non-linear
data.

Exponential model
– y = abx
– For each unit increase in x, y is multiplied
by constant, b.

To transform to linearity, plot log y
against x on the coordinate plane.
Then perform a linear regression.
– log y = a + bx OR ln y = a + bx
Transforming non-linear
data

Power Model
– y = axb
– Often used when trying to use a onedimensional variable (e.g. length), to predict a
multi-dimensional variable (e.g. area, volume,
weight)

To transform to linearity, plot log y against
log x on the coordinate plane. Then
perform a linear regression.
– log y = a + b(log x) OR ln y = a + b(ln x)
Analyzing the relationship
between categorical
variables



A two-way table is used to compare
categorical variables.
Marginal distribution: Analyzing the
totals for one of the variables by itself.
Conditional distribution: The
distribution of the response variable
for each value of the explanatory
variable.