AP Statistics Section 3.1 AScatterplots

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 __________, 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.

tendencies rules

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:

Response:

amount of alcohol consumed

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:

Response:

mean SAT verbal score

mean SAT math score

Note that in first scenario alcohol actually causes a change in body temperature. There is no cause-and-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 __________ ( __ ) axis of a scatterplot. If there is no explanatory-response

distinction, either variable can go on the horizontal axis. Be sure to _______ and

_______ 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.

horizontal X

labelscale

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.__________________________),

form (i.e. ______________) and strength (i.e ____________________________________)

positive or negative association

linear or curved

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 ________

Points hat lie far away from the others in the horizontal direction are ________________

outliers

points linfluentia

NOTE: Always interpret scatterplots in the

context of the problem.

Example 2: Interpret the scatterplot to the right:

Direction:

Form:

Strength:

Outliers:

negative association

slightly curved

fairly strong

Possible outlier at (20, 510)

510). (20,at oputlier possible a is There score.math SATmean

theand SAT the takingstudents ofpercent ebetween th

n associatio negative curvedslightly strong,fairly a is There

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

colony.

join the that birds new ofnumber theandcolony

a return to that birdsadult ofpercent ebetween th

nassociatiolinear negative strong,fairly a is There