More about Correlation

Basketball Players

• A class of 15 students happens to include 5 basketball players.

• True or false, and explain: the relationship between heights and weights for this class should be summarized by r.

Use the Correlation Coefficient?

Nonlinear Association

Exceptional Cases

• The correlation coefficient r is less useful in case there are outliers.

• Moreover, r measures linear association, not association in general.

Ecological Correlation

• In one study, a scatter diagram was shown showing the relationship between the rate of smoking per capita and the rate of deaths from lung cancer in eleven countries.

• The correlation between these eleven pairs of rates was 0.7.

• Comment on this data.

Ecological Correlation

• Correlations based on rates or averages can be misleading.

• For example: in each country, there is a lot of spread around the averages.

• Replacing the people by the country’s average eliminates the spread, and gives the misleading impression of tight clustering.

Example

Suicide and Literacy

• A sociologist is studying the relationship between suicide and literacy in nineteenth-century Italy. He has data for each province. The correlation is 0.6.

• Does this give a fair estimate of the strength of the association between literacy and suicide?

Association is not Causation

• For school children, shoe size is strongly correlated with reading skills.

• In the Great Depression, better educated people tended to have shorter spells of unemployment. Does education protect you against unemployment?

Reading and the TV

• Studies have found a negative association between hours spent watching television and scores on reading tests.

• Does watching television make people less able to read?

Fat in the Diet and Cancer

True or False?

• If r = -0.80, below average values of the dependent variable are associated with below-average values of the independent variable.

• If y is usually less than x, the correlation coefficient between x and y will be negative.

Height and Weight, Again

• An investigator collected data on heights and weights of college students.

• Suppose the correlation coefficient between height and weight for the men was about 0.60; for the women, about the same.

• What happens to the correlation coefficient if you take the men and women together?

Math SAT scores

• ETS computed the average score for each of the 51 states, and the percentage of the high-school seniors in that state who took the test. The correlation between these two variables was equal to –0.86.

• True or false: test scores tend to be lower in the states were a higher percentage take the test.

Math SAT scores

• In New York, the average score was only 471; in Wyoming, the average was 507.

• True or false: the data show that on average, the schools in W. are doing a better job at teaching math than the schools in NY.

Verbal SAT

• ETS computed the average Verbal SAT score for each state, as well as the average Math SAT score. The correlation between these 51 pairs of averages was 0.97. Would the correlation between the Math SAT and Verbal SAT computed from the data on all the individual test takers be larger than 0.97, about 0.97, or less than 0.97?