Topic 2

1. Topic 2 Bivariate Data

2. Bivariate Data • Data for a single variable is univariate data • Many or most real world models have more than one variable … multivariate data • In this topic we will study the relations between two variables … bivariate data

3. Bivariate Data • In many studies, we measure more than one variable for each individual • Some examples are • Rainfall amounts and plant growth • Exercise and cholesterol levels for a group of people • Height and weight for a group of people • In these cases, we are interested in whether the two variables have some kind of a relationship

4. Explanatory and Response Variables • When we have two variables, they could be related in one of several different ways • They could be unrelated • One variable (the explanatory or predictor or independent variable) could be used to explain the other (the response or dependent variable) • One variable could be thought of as causing the other variable to change

5. Explanatory and Response Variables • Sometimes it is not clear which variable is the explanatory variable and which is the response variable • Sometimes the two variables are related without either one being an explanatory variable • Sometimes the two variables are both affected by a third variable called lurking variable that had not been included in the study

6. Examples • Some other examples • Rainfall amounts and plant growth • Explanatory variable – rainfall • Response variable – plant growth • Possible lurking variable – amount of sunlight • Exercise and cholesterol levels • Explanatory variable – amount of exercise • Response variable – cholesterol level • Possible lurking variable – diet

7. Scatter Plot • The most useful graph to show the relationship between two quantitative variables is the scatterplot • Each individual is represented by a point in the diagram • The explanatory (X) variable is plotted on the horizontal scale • The response (Y) variable is plotted on the vertical scale

8. Scatter plot • An example of a scatter plot • Note the truncated vertical scale!

9. Types of Relations • There are several different types of relations between two variables • A relationship is linear when, plotted on a scatter diagram, the points follow the general pattern of a line • A relationship is nonlinear when, plotted on a scatter diagram, the points follow a general pattern, but it is not a line • A relationship has nocorrelation when, plotted on a scatter diagram, the points do not show any pattern

10. Linear Relations • Linear relations have points that cluster around a line • Linear relations can be either positive (the points slants upwards to the right) or negative(the points slant downwards to the right)

11. Positive Linear Relations • For positive (linear) associations • Above average values of one variable are associated with above average values of the other (above/above, the points trend right and upwards) • Below average values of one variable are associated with below average values of the other (below/below, the points trend left and downwards) • Examples • “Age” and “Height” for children • “Temperature” and “Sales of ice cream”

12. Negative Linear Relations • For negative (linear) associations • Above average values of one variable are associated with below average values of the other (above/below, the points trend right and downwards) • Below average values of one variable are associated with above average values of the other (below/above, the points trend left and upwards) • Examples • “Age” and “Time required to run 50 meters” for children • “Temperature” and “Sales of hot chocolate”

13. Nonlinear Relations • Nonlinear relations have points that have a trend, but not around a line • The trend has some bend in it

14. No Relations • When two variables are not related • There is no linear trend • There is no nonlinear trend • Changes in values for one variable do not seem to have any relation with changes in the other

15. ‘Nonlinear Relation’ is not ‘No Relation’ • Nonlinear relations and no relations are very different • Nonlinear relations are definitely patterns … just not patterns that look like lines • No relations are when no patterns appear at all

16. Examples • Examples of nonlinear relations • “Age” and “Height” for people (including both children and adults) • “Temperature” and “Comfort level” for people • Examples of no relations • “Temperature” and “Closing price of the Dow Jones Industrials Index” (probably) • “Age” and “Last digit of telephone number” for adults