Chapter 3
Bivariate Data
Do Tall People Have Big Heads?
• Collect Data
– Enter your height (in inches) and your head circumference (in cm) into
my calculator. Be as exact as possible!
• Graph a scatterplot – label x-axis and y-axis
• Describe the bivariate data
Scatter Plots Vocabulary
Explanatory Variable (x) and Response Variable (y)
Changes in x explain (or even cause) changes in y.
Describe a scatterplot
• Direction: positive or negative
• Form: linear or not (power and exponential in Ch 4)
• Strength: correlation
• Outliers: are there outliers present
Correlation (r)
( measures strength of a scatterplot)
• r is between -1 and 1
•
•
•
•
r = 1 and r = -1 are perfect linear associations
r does not change if you change units (feet to inches, etc)
r ONLY measures LINEAR association
r is not resistant (it is strongly affected by outliers)
±0.6 → 𝑤𝑒𝑎𝑘 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
±0.7 → 𝑚𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
±0.8 → 𝑠𝑡𝑟𝑜𝑛𝑔 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
±0.9 → 𝑣𝑒𝑟𝑦 𝑠𝑡𝑟𝑜𝑛𝑔 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
±1.0 → 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
Least Square Regression Lines
or
Regression Equations
(a.k.a. Line of Best Fit)
Is your 1st term grade in AP stats a good predictor of
your 1st semester grade?
1st Term
61
74
77
64
82
87
95
1st Sem
73
73
85
64
78
85
97
Mrs. Pfeiffer’s AP Stats Class Averages
predicted yˆ
error = observed - predicted
y  yˆ
observed y
Where did it get it’s name?
The sum of all the errors
squared is called the
total sum of squared
errors (SSE).
Calculate the error
(residual) and square it.
Four Key Properties of LSR
 The LSR passes through the point  x , y 
 The LSR sum of residuals (errors) is zero.
 The LSR sum of residuals squared is an absolute
minimum.
 The histogram of the residuals for any value of x has
a normal distribution (as does the histogram of all
the residuals in the LSR)—constant variance.
You MUST know how to Calculate
a Least Squares Linear Regression
Equation using the formulas
LSRL:
Slope:
yˆ  a  bx or yˆ  bo  b1 x
b1  r
sy
sx
Intercept: bo  y  b1 x
Using the output for the graph of the class
averages, answer the following questions:
x  77 . 3 s x  11 . 95
1.
2.
3.
4.
y  79 . 4 s y  10 . 52
r  . 766
2
Write the LSR equation.
Interpret the slope and y-intercept.
What is the value of the correlation coefficient?
If your term grade is 65%, at what percent
would you predict your semester grade?
Interpret SLOPE and Y-INTERCEPT
SLOPE
As x increase by 1, y increases (or decreases) by slope .
Y-INTERCEPT
When x = 0, y is predicted to equal y-intercept .
Extrapolation (pg 203)
Residuals (pg 214)
Coefficient of Determination r2 (pg 223)
Outliers and Influential Points (pg 237)
Lurking Variables (pg 239)
Predicting outside the range of values of the
explanatory variable, x. These predictions are
YEAR
RECORD
typically inaccurate.
Example:
Men’s 800 Meter Run World Records
What reservations you might have
about predicting the record in 2005?
1905
113.4
1915
111.9
1925
111.9
1935
109.7
1945
106.6
1955
105.7
1965
104.3
1975
104.1
1985
101.73
1995
101.73
= observed y – predicted y
= 𝑦−𝑦
To Graph: Plot all points of the form (x, residual)
Good Residual Plot: Scattered (conclude that the
regression line fits the data well)
Bad Residual Plot: Curved or Megaphone (conclude that
the regression line may not be the best model, possibly
a quadratic or exponential function may be more
appropriate)
Look at graphs on pages 216 – 218
Residual
This is exactly what you think it is…the correlation (r) squared.
ALWAYS EXPRESSED AS A PERCENT!
Example 1: Height explains weight. Not totally, but roughly. Suppose r2 is 75% for
a dataset between height and weight. We know that other things affect weight, in
addition to height, including genetics, diet and exercise. So we say that 75% of a
person's variation in weight can be explained by the variation in height, but that
25% of that variation is due to other factors.
Example 2: Suppose you are buying a pizza that is $7 plus $1.50 for each
topping. Clearly, Price = 7 + 1.50(of toppings). Clearly, r and r2 are 1 and
100%. Does this mean that the number of toppings 100% determines my
cost? No, clearly the $7 base price has a lot to do with the price! However, my
variation in price is explained 100% by the variation in the number of toppings I
choose.
How do you INTERPRET it? Use this sentence:
The percent of the variation in y is explained by the linear
relationship between y and x .
Example: 97% of the variation in word record times is explained by
the linear relationship between world record times and the year.
An OUTLIER is an observation that lies outside
the overall pattern of the other observations.
Points can be outliers in the x direction or in
the y direction.
An INFLUENTIAL POINT is an outlier that, if
removed, would significant change the LSRL.
Typically, outliers in the x direction are
influential points.
Child 18 is an outlier in
the x direction.
Child 19 is an outlier in
the y direction.
Child 18 is an influential
point.
Child 19 is not an
influential point.
A LURKING VARIABLE is a variable that is not
among the explanatory or response variables in
the study and yet may influence the interpretation
of relationships among those variables.
Example: Do big feet make you a better speller? Children with
larger shoe sizes in elementary school were found to be better
spellers than their small footed schoolmates. Why?
Association does not imply Causation!
x and y can be associated
a change in x cannot CAUSE a change in y
(unless you have performed a well-designed, well-conducted experiment)