7.1 Seeking Correlation LEARNING GOAL Be able to define correlation, recognize positive and negative correlations on scatter diagrams, and understand the correlation coefficient as a measure of the strength of a correlation. Page 286

Definition A correlation exists between two variables when higher values of one variable consistently go with higher values of another variable or when higher values of one variable consistently go with lower values of another variable. Page 286 Slide 7.1- 2

Here are a few examples of correlations: There is a correlation between the variables amount of smoking and likelihood of lung cancer; that is heavier smokers are more likely to get lung cancer. There is a correlation between the variables height and weight for people; that is, taller people tend to weigh more than shorter people. There is a correlation between the variables demand for apples and price of apples; that is, demand tends to decrease as price increases. There is a correlation between practice time and skill among piano players; that is, those who practice more tend to be more skilled. Page 286 Slide 7.1- 3

Scatter Diagrams Definition A scatter diagram (or scatterplot) is a graph in which each point represents the values of two variables. Pages 286- 287 Slide 7.1- 4

Ice Cream Sales vs Temperature Example: Ice Cream Sales The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days: Ice Cream Sales vs Temperature Temperature °C Ice Cream Sales 14.2° $215 16.4° $325 11.9° $185 15.2° $332 18.5° $406 22.1° $522 19.4° $412 25.1° $614 23.4° $544 18.1° $421 22.6° $445 17.2° $408

Types of Correlation Page 289 Figure 7.3 Types of correlation seen on scatter diagrams. Slide 7.1- 7

Types of Correlation Positive correlation: Both variables tend to increase (or decrease) together. Negative correlation: The two variables tend to change in opposite directions, with one increasing while the other decreases. No correlation: There is no apparent (linear) relationship between the two variables. Nonlinear relationship: The two variables are related, but the relationship results in a scatter diagram that does not follow a straight-line pattern. Page 290 Slide 7.1- 8

Measuring the Strength of a Correlation Statisticians measure the strength of a correlation with a number called the correlation coefficient, represented by the letter r. Page 291 Slide 7.1- 9

Properties of the Correlation Coefficient, r The correlation coefficient, r, is a measure of the strength of a correlation. Its value can range only from -1 to 1. If there is no correlation, the points do not follow any ascending or descending straightline pattern, and the value of r is close to 0. If there is a positive correlation, the correlation coefficient is positive (0 < r ≤ 1): Both variables increase together. A perfect positive correlation (in which all the points on a scatter diagram lie on an ascending straight line) has a correlation coefficient r = 1. Values of r close to 1 mean a strong positive correlation and positive values closer to 0 mean a weak positive correlation. Page 291 Slide 7.1- 10

Properties of the Correlation Coefficient, r (cont,) If there is a negative correlation, the correlation coefficient is negative (-1 ≤ r < 0): When one variable increases, the other decreases. A perfect negative correlation (in which all the points lie on a descending straight line) has a correlation coefficient r = -1. Values of r close to -1 mean a strong negative correlation and negative values closer to 0 mean a weak negative correlation. Page 291 Slide 7.1- 11