Chapter 14 Describing Relationships: Scatterplots and Correlation Chapter 141

2 Thought Question 1 For all cars manufactured in the U.S., there is a positive correlation between the size of the engine and horsepower. There is a negative correlation between the size of the engine and gas mileage. What does it mean for two variables to have a positive correlation or a negative correlation?

Chapter 143 Scatterplot A Scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as the point in the plot fixed by the values of both variables for that individual.

Figure 14.9 Scatterplot of average SAT Mathematics score for each state against the proportion of the state’s high school seniors who took the SAT. The light-colored point corresponds to two states. (This figure was created using the Minitab software package.)

Figure 14.3 Scatterplot of the life expectancy of people in many nations against each nation’s gross domestic product per person. (This figure was created using the Minitab software package.)

Chapter 146 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.

Chapter 147 Positive association, Negative association Two variables are positively/Negatively associated when above-average values of one tend to accompany above-average /below-average values of the other. The scatterplot slops upward/downward as we move from the left to right.

Our scatterplot regarding the SAT scores shows two clusters of states. The one with the GDP shows a curved relationship. The strength of a relationship in a scatterplot is determined by how closely the points follow a clear form. The relationship in both our plots are not strong. – States with similar percentages show quite a bit of scatter in their average scores. – Nations with similar GDPs can have quite different life expectancies. Chapter 148

A scatterplot with strong relationship Chapter 149 Figure 14.5 Scatterplot of the lengths of two bones in 5 fossil specimens of the extinct beast Archaeopteryx.

Statistical versus Deterministic Relationships Distance versus Speed (when travel time is constant). Income (in millions of dollars) versus total assets of banks (in billions of dollars). Chapter 1410

Distance versus Speed Distance = Speed  Time Suppose time = 1.5 hours Each subject drives a fixed speed for the 1.5 hrs. – speed chosen for each subject varies from 10 mph to 50 mph Distance does not vary for those who drive the same fixed speed Deterministic relationship Chapter 1411

Income versus Assets Income = a + b  Assets? Assets vary from 3.4 billion to 49 billion Income varies from bank to bank, even among those with similar assets Statistical relationship Chapter 1412

Linear Relationship Some relationships are such that the points of a scatterplot tend to fall along a straight line -- linear relationship. Chapter 1413

Measuring Strength & Direction of a Linear Relationship How closely does a non-horizontal straight line fit the points of a scatterplot? The correlation coefficient (often referred to as just correlation) r is a measure of: – the strength of the relationship: the stronger the relationship, the larger the magnitude of r. – the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship. Chapter 1414

Correlation Coefficient special values for r :  A perfect positive linear relationship would have r = +1.  A perfect negative linear relationship would have r = -1.  If there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0.  Note: r must be between -1 and +1, inclusive. r > 0: as one variable changes, the other variable tends to change in the same direction. r < 0: as one variable changes, the other variable tends to change in the opposite direction. Chapter 1415

Figure 14.7 How correlation measures the strength of a straight-line relationship. Patterns closer to a straight line have correlations closer to 1 or −1.

Chapter 1417 Correlation Calculation Suppose we have data on variables X and Y for n individuals: x 1, x 2, …, x n and y 1, y 2, …, y n Each variable has a mean and std dev:

Chapter 1418 Case Study Per Capita Gross Domestic Product and Average Life Expectancy for Countries in Western Europe

Chapter 1419 Case Study CountryPer Capita GDP (x)Life Expectancy (y) Austria Belgium Finland France Germany Ireland Italy Netherlands Switzerland United Kingdom

Chapter 1420 Case Study xy = = sum = s x =1.532s y =0.795

Chapter 1421 Case Study There is a strong, positive linear relationship between Per Capita GDP (x) and Life Expectancy (y).

Problems with Correlations Outliers can inflate or deflate correlations. Groups combined inappropriately may mask relationships (a third variable). – groups may have different relationships when separated. Chapter 1422

Figure 14.8 Moving one point reduces the correlation from r = to r =

Not all Relationships are Linear Miles per Gallon versus Speed Linear relationship? MPG = a + b  Speed? Speed chosen for each subject varies from 20 mph to 60 mph. MPG varies from trial to trial, even at the same speed. Statistical relationship Chapter 1424

Not all Relationships are Linear Miles per Gallon versus Speed Curved relationship (r is misleading) Speed chosen for each subject varies from 20 mph to 60 mph. MPG varies from trial to trial, even at the same speed. Statistical relationship Chapter 1425

Price of Books versus Size Relationship between price of books and the number of pages? Positive? Look at paperbacks: Look at hardcovers: All books together: Overall correlation is Negative! Chapter 1426

Key Concepts Statistical vs. Deterministic Relationships Statistically Significant Relationship Strength of Linear Relationship Direction of Linear Relationship Correlation Coefficient Problems with Correlations Chapter 1427