Reminders/Comments. Thanks for the quick feedback I ll try to put HW up on Saturday and I ll you

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1 Reminders/Comments Thanks for the quick feedback I ll try to put HW up on Saturday and I ll you Final project will be assigned in the last week of class You ll have that week to do it Participation is awarded for attendance, vocal and group participation in lecture and quiz section, and discussion board HW due and quiz to take tomorrow 1

2 Warm Up In the mid-19th century, Francis Galton wanted to know whether fathers pass some physical characteristics on to their sons. To investigate this question, he traveled around Britain and recorded the heights of 1000 first born sons and the heights of their fathers. He found that tall fathers tend to have tall sons, and short fathers tend to have short sons. What are the explanatory and response variables? Does this study give evidence of a causal relationship? Why or why not? 2

3 Chapter 14: Describing Relationships (Scatterplots and Correlation) Aaron Zimmerman STAT Summer 2014 Department of Statistics University of Washington - Seattle 3

4 Motivating Example Francis Galton s study is an example of a multivariate study Most studies examine data on more than one variable Our strategy for multivariate problems is the same 1) First plot the data, then add numerical summaries 2) Look for overall patterns and deviations from those patterns 3) When the overall pattern is quite regular, there is sometimes a way to describe it concisely 4

5 Motivating Example The most common graphical way to describe the relation between two quantitative variables is a scatterplot The explanatory variable (if there is one) goes on the x-axis The response variable (if there is one) goes on the y-axis 5

6 Motivating Example As with univariate graphs, the first step is to look for the overall pattern and for striking deviations from that pattern You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship An important kind of deviation is an outlier, an individual value that falls outside the overall pattern of the relationship 6

7 Motivating Example The direction of a scatterplot is described by the association between the variables Two variables are positively associated when above average values of one tend to accompany above average values of the other and vice versa. The scatterplot slopes upward as we move from left to right Heights of fathers are positively associated with heights of sons 7

8 The direction of a scatterplot is described by the association between the variables Two variables are negatively associated when above average values of one tend to accompany below average values of the other and vice versa. The scatterplot slopes downward as we move from left to right Motivating Example Key idea #1: If there is strong association between two variables, knowing one helps a lot in predicting the other. But when there is a weak association, information about one variable does not help much in guessing the other. 8

9 Motivating Example The form of a scatterplot can be linear or curved The relationship between heights of fathers and heights of sons is roughly linear The strength of a relationship is determined by how closely the points follow a clear form The relationship between heights of father and heights of sons is moderately strong 9

10 Scatterplot Practice Was the average midterm score around 25, 50, or 75? Was the SD of the midterm scores around 5, 10, or 20? Was the SD of the final scores around 5, 10, or 20? Which exam was harder? Was there more spread in the midterm or final scores? T/F: There was a strong positive association between midterm and final scores 10

11 Motivating Example There are a few outliers in Galton s scatterplot Some short fathers had very tall sons Some relatively tall fathers had very short sons For correlation and regression (the next two topics), it will be important to note whether there are many outliers in the same direction It doesn t look like there are any more outliers in one direction than another 11

12 Correlation When the relationship between two variables is roughly linear, the direction and strength of the relationship can be described numerically by the correlation r Positive r indicates positive association between the variables, while negative r indicates negative association The correlation r always falls between -1 and 1 Values of r near zero indicate a very weak straight-line relationship The strength of the relationship increases as r moves away from 0 towards either -1 or 1. Values of r close to -1 or 1 indicate that the points are almost in a straight line 12

13 Correlation 13

14 Correlation The correlation between heights of fathers and heights of sons must be positive because the variables are positively associated The exact value is r = 0.52 Again, this indicates a moderately positive linear relationship between the variables 14

15 Four other reminders about correlation The correlation between two variables does not change when we change the units of measurement Correlation ignores the distinction between explanatory and response variables The correlation is strongly affected by a few outliers Correlation only measures linear relationships Association can be used to describe any relationship Correlation only measures the direction and strength of linear relationships 15

16 United Nations Data Example There is a great deal of interest in identifying variables that are associated with the average life expectancy in countries around the world The following three examples will examine three potential explanatory variables The percent of the population who attends primary school The log per capita GDP of the country The annual population growth in the country In each example, the response variable is the average life expectancy in the country 16

17 Example #1 Can we use correlation to describe the relationship? How would you describe the correlations direction? How would you describe the correlations strength? What is your estimate of the value of the correlation? 17

18 Example #1 The association between primary school attendance and life expectancy is positive, but the form is curved So, it is not appropriate to use correlation as a numerical summary of this relationship 18

19 Example #2 Can we use correlation to describe the relationship? How would you describe the correlations direction? How would you describe the correlations strength? What is your estimate of the value of the correlation? 19

20 Example #2 We can use correlation, it looks more or less linear I d say the correlation is strong and positive The correlation is around

21 Example #3 Can we use correlation to describe the relationship? How would you describe the correlations direction? How would you describe the correlations strength? What is your estimate of the value of the correlation? 21

22 Example #3 We can use correlation, it looks more or less linear I d say the correlation is moderate and negative The correlation is around

23 Describe the correlation For social scientists Sociologists at the University of Virginia (Oishi et al 2011) investigated the association between the level of income inequality in the U.S. each year since 1972 and the level of happiness in that year. They calculated the Gini coefficient (a measure of income inequality) in each year, and compared these values with average survey responses in each year to the question, how happy are you? For other scientists River ecologists at the American Museum of Natural History investigated the association between temperature and oxygen levels in river water. They measured the water temperature (in degrees Celcius) at 500 locations, and compared these values to the number of mg/l of dissolved oxygen in the water at each location. 23

24 Describe the correlation 24

25 Describe the correlation For social scientists Weak, negative correlation r = 0.32 For other scientists Strong negative correlation r =

26 Homework Read Chapter 14 Do problems 14.6 (explain why), 14.7, 14.13, 14.17, 14.19, 14.22, (also report the correlation between the variables) 26

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