Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 1 of 30 Chapter 4 Section 1 Scatter Diagrams and Correlation

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 2 of 30 Chapter 4 – Section 1 ●Learning objectives  Draw and interpret scatter diagrams  Understand the properties of the linear correlation coefficient  Compute and interpret the linear correlation coefficient  Determine whether there is a linear relation between two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 3 of 30 Chapter 4 – Section 1 ●Learning objectives  Draw and interpret scatter diagrams  Understand the properties of the linear correlation coefficient  Compute and interpret the linear correlation coefficient  Determine whether there is a linear relation between two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 4 of 30 Chapter 4 – Section 1 ●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 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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 5 of 30 Chapter 4 – Section 1 ●When we have two variables, they could be related in one of several different ways  They could be unrelated ●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 variable) could be used to explain the other (the response or dependent variable) ●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 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 ●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 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 ●In this chapter, we examine the second case … explanatory and response variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 6 of 30 Chapter 4 – Section 1 ●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, a lurking variable, that had not been included in the study

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 7 of 30 Chapter 4 – Section 1 ●An example of a lurking variable ●A researcher studies a group of elementary school children  Y = the student’s height  X = the student’s shoe size ●An example of a lurking variable ●A researcher studies a group of elementary school children  Y = the student’s height  X = the student’s shoe size ●It is not reasonable to claim that shoe size causes height to change ●An example of a lurking variable ●A researcher studies a group of elementary school children  Y = the student’s height  X = the student’s shoe size ●It is not reasonable to claim that shoe size causes height to change ●The lurking variable of age affects both of these two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 8 of 30 Chapter 4 – Section 1 ●Some other examples ●Rainfall amounts and plant growth  Explanatory variable – rainfall  Response variable – plant growth  Possible lurking variable – amount of sunlight ●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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 9 of 30 Chapter 4 – Section 1 ●The most useful graph to show the relationship between two quantitative variables is the scatter diagram ●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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 10 of 30 Chapter 4 – Section 1 ●An example of a scatter diagram ●Note the truncated vertical scale!

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 11 of 30 Chapter 4 – Section 1 ●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 ●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 ●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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 12 of 30 Chapter 4 – Section 1 ●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)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 13 of 30 Chapter 4 – Section 1 ●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) ●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”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 14 of 30 Chapter 4 – Section 1 ●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) ●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”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 15 of 30 Chapter 4 – Section 1 ●Nonlinear relations have points that have a trend, but not around a line ●The trend has some bend in it

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 16 of 30 Chapter 4 – Section 1 ●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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 17 of 30 Chapter 4 – Section 1 ●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 ●This distinction will be very important in the remainder of this chapter

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 18 of 30 Chapter 4 – Section 1 ●Examples of nonlinear relations  “Age” and “Height” for people (including both children and adults)  “Temperature” and “Comfort level” for people ●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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 19 of 30 Chapter 4 – Section 1 ●Learning objectives  Draw and interpret scatter diagrams  Understand the properties of the linear correlation coefficient  Compute and interpret the linear correlation coefficient  Determine whether there is a linear relation between two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 20 of 30 Chapter 4 – Section 1 ●The linear correlation coefficient is a measure of the strength of linear relation between two quantitative variables ●The sample correlation coefficient “r” is ●This should be computed with software (and not by hand) whenever possible

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 21 of 30 Chapter 4 – Section 1 ●Some properties of the linear correlation coefficient  r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms) ●Some properties of the linear correlation coefficient  r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms)  r is always between –1 and +1 ●Some properties of the linear correlation coefficient  r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms)  r is always between –1 and +1  Positive values of r correspond to positive relations ●Some properties of the linear correlation coefficient  r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms)  r is always between –1 and +1  Positive values of r correspond to positive relations  Negative values of r correspond to negative relations

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 22 of 30 Chapter 4 – Section 1 ●Some more properties of the linear correlation coefficient  The closer r is to +1, the stronger the positive relation … when r = +1, there is a perfect positive relation ●Some more properties of the linear correlation coefficient  The closer r is to +1, the stronger the positive relation … when r = +1, there is a perfect positive relation  The closer r is to –1, the stronger the negative relation … when r = –1, there is a perfect negative relation ●Some more properties of the linear correlation coefficient  The closer r is to +1, the stronger the positive relation … when r = +1, there is a perfect positive relation  The closer r is to –1, the stronger the negative relation … when r = –1, there is a perfect negative relation  The closer r is to 0, the less of a linear relation (either positive or negative)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 23 of 30 Chapter 4 – Section 1 ●Examples of positive correlation Strong Positive r =.8 Moderate Positive r =.5 Very Weak r =.1 ●Examples of positive correlation ●In general, if the correlation is visible to the eye, then it is likely to be strong

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 24 of 30 Chapter 4 – Section 1 ●Examples of negative correlation Strong Negative r = –.8 Moderate Negative r = –.5 Very Weak r = –.1 ●Examples of negative correlation ●In general, if the correlation is visible to the eye, then it is likely to be strong

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 25 of 30 Chapter 4 – Section 1 ●Nonlinear correlation and no correlation Nonlinear RelationNo Relation ●Nonlinear correlation and no correlation ●Both sets of variables have r = 0.1, but the difference is that the nonlinear relation shows a clear pattern

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 26 of 30 Chapter 4 – Section 1 ●Learning objectives  Draw and interpret scatter diagrams  Understand the properties of the linear correlation coefficient  Compute and interpret the linear correlation coefficient  Determine whether there is a linear relation between two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 27 of 30 Chapter 4 – Section 1 ●Correlation is not causation! ●Just because two variables are correlated does not mean that one causes the other to change ●Correlation is not causation! ●Just because two variables are correlated does not mean that one causes the other to change ●There is a strong correlation between shoe sizes and vocabulary sizes for grade school children  Clearly larger shoe sizes do not cause larger vocabularies  Clearly larger vocabularies do not cause larger shoe sizes ●Correlation is not causation! ●Just because two variables are correlated does not mean that one causes the other to change ●There is a strong correlation between shoe sizes and vocabulary sizes for grade school children  Clearly larger shoe sizes do not cause larger vocabularies  Clearly larger vocabularies do not cause larger shoe sizes ●Often lurking variables result in confounding

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 28 of 30 Chapter 4 – Section 1 ●Learning objectives  Draw and interpret scatter diagrams  Understand the properties of the linear correlation coefficient  Compute and interpret the linear correlation coefficient  Determine whether there is a linear relation between two variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 29 of 30 Chapter 4 – Section 1 ●How large does the correlation coefficient have to be before we can say that there is a relation? ●We’re not quite ready to answer that question (that’s Chapter 12 – Section 3) ●For now, we can look at Table VIII in Appendix A ●How large does the correlation coefficient have to be before we can say that there is a relation? ●We’re not quite ready to answer that question (that’s Chapter 12 – Section 3) ●For now, we can look at Table VIII in Appendix A ●For example for n = 15  A correlation coefficient of greater than would indicate a positive linear correlation  A correlation coefficient of less than –0.514 would indicate a negative linear correlation

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 1 – Slide 30 of 30 Summary: Chapter 4 – Section 1 ●Correlation between two variables can be described with both visual and numeric methods ●Visual methods  Scatter diagrams  Analogous to histograms for single variables ●Numeric methods  Linear correlation coefficient  Analogous to mean and variance for single variables ●Care should be taken in the interpretation of linear correlation (nonlinearity and causation)