INTERPRETING SCATTERPLOTS

The researchers believe that lean body mass is an important influence on metabolic rate. Make a scatterplot to examine this belief. (The Two-Variable Statistical Calcu-lator Applet provides an easy way to make scatterplots. Click “Data” to enter your data, then “Scatterplot” to see the plot.)

4.5 Outsourcing by airlines. Airlines have increasingly outsourced the maintenance of their planes to other companies. A concern voiced by critics is that the maintenance may be less carefully done, so that outsourcing creates a safety hazard. In addition, flight delays are often due to maintenance problems, so one might look at government data on percent of major maintenance outsourced and percent of flight delays blamed on the airline to determine if these concerns are justified. This was done, and data from 2005 and 2006 appeared to justify the concerns of the critics. Do more recent data still support the concerns of the critics? Here are data from 2009:³ AIRLINES

Outsource Delay Outsource Delay Airline percent percent Airline percent percent

AirTran 52.8 24.21 Hawaiian 74.1 7.94

Alaska 56.8 17.09 JetBlue 53.7 22.55

American 23.3 22.71 Northwest 59.8 20.85

Continental 44.5 21.23 Southwest 61.7 17.00

Delta 25.6 21.44 United 40.6 19.02

Frontier 26.8 21.70 US Airways 60.4 19.13

Make a scatterplot that shows the relation between delays and outsourcing.

INTERPRETING SCATTERPLOTS

To interpret a scatterplot, adapt the strategies of data analysis learned in Chapters 1 and 2 to the new two-variable setting.

Mass 36.1 54.6 48.5 42.0 50.6 42.0 40.3 33.1 42.4 34.5 51.1 41.2 Rate 995 1425 1396 1418 1502 1256 1189 913 1124 1052 1347 1204

EXAMINING A SCATTERPLOT

In any graph of data, look for the overall pattern and for striking deviations from that pattern.

You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship.

An important kind of deviation is an outlier, an individual value that falls outside the overall pattern of the relationship.

weight leaving out all fat. Metabolic rate is measured in calories burned per 24 hours, the same calories used to describe the energy content of foods. METABOLIC

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E X A M P L E 4 . 4

We continue to explore the state SAT Mathematics scores by interpreting what the scatterplot tells us about the variation in scores from state to state.

SOLVE (INTERPRET THE PLOT): Figure 4.1 shows a clear direction: the overall pattern moves from upper left to lower right. That is, states in which higher percents of high school graduates take the SAT tend to have lower mean SAT Mathematics scores. We call this a negative association between the two variables.

The form of the relationship is roughly a straight line with a slight curve to the right as it moves down. What is more, most states fall into two distinct clusters. As in the histogram in Figure 1.8, the ACT states cluster at the left and the SAT states at the right. In 22 states, fewer than 20% of seniors took the SAT; in another 22 states, more than 50% took the SAT.

The strength of a relationship in a scatterplot is determined by how closely the points follow a clear form. The overall relationship in Figure 4.1 is moderately strong:

states with similar percents taking the SAT tend to have roughly similar mean SAT Math scores.

CONCLUDE: Percent taking explains much of the variation among states in average SAT Mathematics score. States in which a higher percent of students take the SAT tend to have lower mean scores because the mean includes a broader group of students.

SAT states as a group have lower mean SAT scores than ACT states. So average SAT score says almost nothing about the quality of education in a state. It is foolish to

“rank” states by their average SAT scores. ■

clusters

POSITIVE ASSOCIATION, NEGATIVE ASSOCIATION

Two variables are positively associated when above-average values of one tend to accompany above-average values of the other, and below-average values also tend to occur together.

Two variables are negatively associated when above-average values of one tend to accompany below-average values of the other, and vice versa.

Of course, not all relationships have a clear direction that we can describe as positive association or negative association. Exercise 4.8 gives an example that does not have a single direction. Here is an example of a strong positive associa-tion with a simple and important form.

E X A M P L E 4 . 5

STATE: Manatees are large, gentle, slow-moving creatures found along the coast of Florida. Many manatees are injured or killed by boats. Table 4.1 contains data on the number of boats registered in Florida (in thousands) and the number of manatees killed by boats for the years between 1977 and 2009.⁴ Examine the relationship. Is it plausible that restricting the number of boats would help protect manatees?

Douglas Faulkner/Photo Researchers

1 0 2 C H A P T E R 4

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T A B L E 4 . 1 Florida boat registrations (thousands) and manatees killed by boats

YEAR BOATS MANATEES YEAR BOATS MANATEES YEAR BOATS MANATEES

1977 447 13 1989 711 50 2001 944 81

1978 460 21 1990 719 47 2002 962 95

1979 481 24 1991 681 53 2003 978 73

1980 498 16 1992 679 38 2004 983 69

1981 513 24 1993 678 35 2005 1010 79

1982 512 20 1994 696 49 2006 1024 92

1983 526 15 1995 713 42 2007 1027 73

1984 559 34 1996 732 60 2008 1010 90

1985 585 33 1997 755 54 2009 982 97

1986 614 33 1998 809 66

1987 645 39 1999 830 82

1988 675 43 2000 880 78

PLAN: Make a scatterplot with “boats registered” as the explanatory variable and

“manatees killed” as the response variable. Describe the form, direction, and strength of the relationship.

SOLVE: Figure 4.2 is the scatterplot. There is a positive association—more boats goes with more manatees killed. The form of the relationship is linear. That is, the overall pattern follows a straight line from lower left to upper right. The relationship is strong because the points don’t deviate greatly from a line.

linear relationship

100809070604050302010

400 500 600 700 800 900 1000 1100

Boats registered in Florida (thousands)

Florida manatees killed by boats

This scatterplot has a linear (straight-line) overall pattern.

F I G U R E 4 . 2

Scatterplot of the number of Florida manatees killed by boats in the years 1977 to 2009 against the number of boats registered in Florida that year, for Example 4.5. There is a strong linear (straight-line) pattern.

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CONCLUDE: As more boats are registered, the number of manatees killed by boats goes up linearly. Data from the Florida Wildlife Commission indicate that in recent years boats accounted for 24% of manatee deaths and 31% of deaths whose causes could be determined. Although many manatees die from other causes, it appears that fewer boats would mean fewer manatee deaths. ■

As the following chapter will emphasize, it is wise to always ask what other variables lurking in the background might contribute to the relationship displayed in a scatterplot. Because both boats registered and manatees killed are recorded year by year, any change in conditions over time might affect the relationship. For example, if boats in Florida have tended to go faster over the years, that might result in more manatees killed by the same number of boats.

A P P LY Y O U R K N O W L E D G E

4.6 Do heavier people burn more energy? Describe the direction, form, and strength of the relationship between lean body mass and metabolic rate, as displayed in your plot for Exercise 4.4. METABOLIC

4.7 Outsourcing by airlines. Does your plot for Exercise 4.5 show a positive, nega-tive, or no association between maintenance outsourcing and delays caused by the airline? One airline is a low outlier in delay percent. Which airline is this? Aside from the outlier, does the plot show a roughly linear form? If it does, is the relation-ship very strong? AIRLINES

4.8 Does fast driving waste fuel? How does the fuel consumption of a car change as its speed increases? Here are data for a British Ford Escort. Speed is measured in kilometers per hour, and fuel consumption is measured in liters of gasoline used per 100 kilometers traveled.⁵ FASTDRIVE

Speed 10 20 30 40 50 60 70 80

Fuel 21.00 13.00 10.00 8.00 7.00 5.90 6.30 6.95 Speed 90 100 110 120 130 140 150 Fuel 7.57 8.27 9.03 9.87 10.79 11.77 12.83 (a) Make a scatterplot. (Which is the explanatory variable?)

(b) Describe the form of the relationship. It is not linear. Explain why the form of the relationship makes sense.

(c) It does not make sense to describe the variables as either positively associated or negatively associated. Why?

(d) Is the relationship reasonably strong or quite weak? Explain your answer.

ADDING CATEGORICAL VARIABLES