Topic 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

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

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

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

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

Scatter Plot The most useful graph to show the relationship between two quantitative variables is the scatter plot 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

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

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 no correlation when, plotted on a scatter diagram, the points do not show any pattern

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)

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”

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”

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

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

‘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

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