AP Statistics Section 3.1 A
Scatterplots
A medical study finds that short women
are more likely to have heart attacks than
women of average height, while tall
women have the fewest heart attacks. An
insurance group reports that heavier cars
have fewer deaths per 100,000 vehicles
than do lighter cars. These and many other
statistical studies look at the relationship
between two variables.
CAUTION: Statistical relationships are
__________,
rules
tendencies not ______.
They allow individual exceptions. For
example, although smokers, on average,
die younger than nonsmokers, some
people live to 90 while smoking three
packs a day.
To understand a statistical relationship between
two variables, we measure both variables on the
same individual.
When analyzing a relationship between two
variables, we must examine other variables as
well. Name two variables that could affect the
heart attack study above.
weight, exersize, stress, heredity
Researchers need to eliminate the
effect of these other variables. One
of our main themes is that the
relationship between two
variables can be strongly
influenced by other variables
lurking in the background.
In chapter 3, we will focus on
relationships between quantitative
variables. Categorical variables will
be examined in chapter 4. Quite
often, you want to determine if
one of the variables helps explain
or even causes changes in the
other.
A response variable measures an
outcome of a study.
An explanatory variable helps
explain or influences the changes
in the response variable.
Explanatory variables are
sometimes called independent
variables and response variables
are sometimes called dependent
variables.
Example 1: Identify the explanatory
and response variable in the
following scenarios:
Alcohol has many effects on the body. One effect is a
drop in body temperature. To study this effect,
researchers give several different amount of alcohol
to mice, then measure the change in each mouse’s
body temperature in the 15 minutes after taking the
alcohol.
Explanatory: amount of alcohol consumed
Response: change in body temperature
Jim wants to know how the mean 2005 SAT Math and
Verbal scores in the 50 states are related to each
other. He wants to determine if he can predict a
state’s mean SAT Math score if he knows the mean
SAT Verbal score.
Explanatory: mean SAT verbal score
Response: mean SAT math score
Note that in first scenario alcohol
actually causes a change in body
temperature. There is no causeand-effect relationship between
SAT Verbal and Math scores.
Caution: Calling one variable
explanatory and the other
responsive doesn’t necessarily
mean that changes in one cause
changes in the other.
The most effective way to display the
relationship between two quantitative variables
is with a scatterplot. To draw a scatterplot by
hand: Plot the explanatory variable, if there is
one, on the __________
horizontal ( __
X ) axis of a
scatterplot. If there is no explanatory-response
distinction, either variable can go on the
label and
horizontal axis. Be sure to _______
_______
scale both axes. The intervals on each axis
must be uniform for that axis; and adopt a scale
that uses the whole grid and allows the details
to be easily seen.
Two variables are positively
associated when above average
values tend to occur together on
both variables.
Two variables are negatively
associated when above average
values on one variable tend to
occur with below average values
on the other.
When describing the overall pattern of a scatterplot you
must address 3 key areas:
direction (i.e.__________________________),
positive or negative association
linear or curved and
form (i.e. ______________)
strength (i.e ____________________________________)
how closely do the points follow the form
Also note unusual aspects such as clusters
of points or points that lie far outside the
pattern.
Points that lie far away from all the others
in the vertical direction are called
________
outliers
Points hat lie far away from the others in
the horizontal direction are
________________
influential points
NOTE: Always interpret
scatterplots in the
context of the problem.
Example 2: Interpret the scatterplot to the right:
Direction: negative association
Form: slightly curved
Strength: fairly strong
Outliers: Possible outlier at (20, 510)
There is a fairly strong, slightly curved negative association
between the percent of students taking the SAT and the
mean SAT math score. There is a possible oputlier at (20, 510).
To add a categorical variable to a
scatterplot, use a different color or
symbol for each category.
Scatterplots on the TI 83/84:
1. Put the data into two lists
2. STATPLOT (2nd func. of Y=)
3. Make sure all but the first are turned OFF
4. Turn plot1 ON and highlight the first graph and
press ENTER
5. Xlist is the explanatory variable and Ylist is the
response variable
6. Choose the “marker” you wish to use
7. Press GRAPH and ZOOM 9 for appropriate window
Example: How does the percent of adult birds in a colony
from one year that return to nest the following year,
affect the number of new birds that join the colony? Here
are data for 13 colonies of sparrowhawks. Construct a
scatterplot and describe what you see.
% returning: 74 66 81 52 73 62 52 45 62 46 60 46 38
# new birds: 5 6 8 11 12 15 16 17 18 18 19 20 20
There is a fairly strong, negative linear association
between the percent of adult birds that return to a
colony and the number of new birds that join the
colony.