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Chapter 7 Scatterplots and Correlation Chapter 7 Objectives Scatterplots Scatterplots Explanatory and response variables Interpreting scatterplots Outliers Categorical variables in scatterplots Basic Terminology Univariate data: 1 variable is measured on each sample unit or population unit (chapters 2 through 6) e.g. height of each student in a sample Bivariate data: 2 variables are measured on each sample unit or population unit e.g. height and GPA of each student in a sample; (caution: data from 2 separate samples is not bivariate data) Basic Terminology (cont.) Multivariate data: several variables are measured on each unit in a sample or population. For each student in a sample of NCSU students, measure height, GPA, and distance between NCSU and hometown; Focus on bivariate data in chapter 7 Same goals with bivariate data that we had with univariate data Graphical displays and numerical summaries Seek overall patterns and deviations from those patterns Descriptive measures of specific aspects of the data Here, we have two quantitative variables for each of 16 students. 1) How many beers they drank, and 2) Their blood alcohol level (BAC) We are interested in the relationship between the two variables: How is one affected by changes in the other one? Student Beers Blood Alcohol 1 5 0.1 2 2 0.03 3 9 0.19 4 7 0.095 5 3 0.07 6 3 0.02 7 4 0.07 8 5 0.085 9 8 0.12 10 3 0.04 11 5 0.06 12 5 0.05 13 6 0.1 14 7 0.09 15 1 0.01 16 4 0.05 Scatterplots Useful method to graphically describe the relationship between 2 quantitative variables Scatterplot: Blood Alcohol Content vs Number of Beers In a scatterplot, one axis is used to represent each of the variables, and the data are plotted as points on the graph. Student Beers BAC 1 5 0.1 2 2 0.03 3 9 0.19 4 7 0.095 5 3 0.07 6 3 0.02 7 4 0.07 8 5 0.085 9 8 0.12 10 3 0.04 11 5 0.06 12 5 0.05 13 6 0.1 14 7 0.09 15 1 0.01 16 4 0.05 Focus on Three Features of a Scatterplot Look for an overall pattern regarding … 1. Shape - ? Approximately linear, curved, up-and-down? 2. Direction - ? Positive, negative, none? 3. Strength - ? Are the points tightly clustered in the particular shape, or are they spread out? Blood Alcohol as a function of Number of Beers … and deviations from the overall pattern: Outliers Blood Alcohol Level (mg/ml) 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 1 2 3 4 5 6 Number of Beers 7 8 9 10 Scatterplot: Fuel Consumption vs Car Weight. x=car weight, y=fuel cons. (xi, yi): (3.4, 5.5) (3.8, 5.9) (4.1, 6.5) (2.2, 3.3) (2.6, 3.6) (2.9, 4.6) (2, 2.9) (2.7, 3.6) (1.9, 3.1) (3.4, 4.9) FUEL CONSUMP. (gal/100 miles) FUEL CONSUMPTION vs CAR WEIGHT 7 6 5 4 3 2 1.5 2.5 3.5 WEIGHT (1000 lbs) 4.5 Explanatory and response variables A response variable measures or records an outcome of a study. An explanatory variable explains changes in the response variable. Typically, the explanatory or independent variable is plotted on the x axis, and the response or dependent variable is plotted on the y axis. Blood Alcohol as a function of Number of Beers Blood Alcohol Level (mg/ml) 0.20 Response (dependent) variable: blood alcohol content y 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 x 0 1 2 3 4 5 6 7 8 9 10 Number of Beers Explanatory (independent) variable: number of beers SAT Score vs Proportion of Seniors Taking SAT 2005 2005 Average SAT Score 2005 SAT Total 1250 IW IL 1200 1150 NC 74% 1010 1100 1050 1000 DC 950 0% 20% 40% 60% Percent of Seniors Taking SAT 80% 100% Some plots don’t have clear explanatory and response variables. Do calories explain sodium amounts? Does percent return on Treasury bills explain percent return on common stocks? Making Scatterplots Excel: In text: see p. 169-170 Statcrunch In the left panel of our class webpage http://www.stat.ncsu.edu/people/reiland/courses/st311/ click on Student Resources, in “Statcrunch Instructional Videos” see “Scatterplots and Regression” instructional video; in “Many Statcrunch Instructional Videos” see videos 15 and 47. TI calculator: In the left panel of our class web page click on Student Resources, under “Graphing Calculators, Online Calculations” click on “TI Graphing Calculator Guide”; see p. 7-9. Form and direction of an association Linear No relationship Nonlinear Positive association: High values of one variable tend to occur together with high values of the other variable. Negative association: High values of one variable tend to occur together with low values of the other variable. No relationship: X and Y vary independently. Knowing X tells you nothing about Y. One way to think about this is to remember the following: The equation for this line is y = 5. x is not involved. Strength of the association The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form. With a strong relationship, you can get a pretty good estimate of y if you know x. With a weak relationship, for any x you might get a wide range of y values. This is a weak relationship. For a particular state median household income, you can’t predict the state per capita income very well. This is a very strong relationship. The daily amount of gas consumed can be predicted quite accurately for a given temperature value. How to scale a scatterplot Same data in all four plots Using an inappropriate scale for a scatterplot can give an incorrect impression. Both variables should be given a similar amount of space: • Plot roughly square • Points should occupy all the plot space (no blank space) Outliers An outlier is a data value that has a very low probability of occurrence (i.e., it is unusual or unexpected). In a scatterplot, outliers are points that fall outside of the overall pattern of the relationship. Not an outlier: Outliers The upper right-hand point here is not an outlier of the relationship—It is what you would expect for this many beers given the linear relationship between beers/weight and blood alcohol. This point is not in line with the others, so it is an outlier of the relationship. IQ score and Grade point average a) Describe in words what this plot shows. b) Describe the direction, shape, and strength. Are there outliers? c) What is the deal with these people? Categorical variables in scatterplots Often, things are not simple and one-dimensional. We need to group the data into categories to reveal trends. What may look like a positive linear relationship is in fact a series of negative linear associations. Plotting different habitats in different colors allows us to make that important distinction. Comparison of men and women racing records over time. Each group shows a very strong negative linear relationship that would not be apparent without the gender categorization. Relationship between lean body mass and metabolic rate in men and women. Both men and women follow the same positive linear trend, but women show a stronger association. As a group, males typically have larger values for both variables. Correlation Objectives Correlation The correlation coefficient “r” r does not distinguish x and y r has no units r ranges from -1 to +1 Influential points The correlation coefficient "r" The correlation coefficient is a measure of the direction and strength of the linear relationship between 2 quantitative variables. It is calculated using the mean and the standard deviation of both the x and y variables. Time to swim: x = 35, sx = 0.7 Pulse rate: y = 140 sy = 9.5 Correlation can only be used to describe quantitative variables. Categorical variables don’t have means and standard deviations. Part of the calculation involves finding z, the standardized score we used when working with the normal distribution. You DON'T want to do this by hand. Make sure you learn how to use your calculator! Calculating Correlation Excel: In text: see p. 169 (bottom of second column in Excel section) Statcrunch In the left panel of our class webpage http://www.stat.ncsu.edu/people/reiland/courses/st311/ click on Student Resources, in “Many Statcrunch Instructional Videos” see videos 18 and 22. TI calculator: In the left panel of our class webpage click on Student Resources; under “Graphing Calculators, Online Calculations”, either click on TI Graphing Calculator Guide and see p. 8, or Click on Online Graphing Calculator Tutorials Example: calculating correlation (x1, y1), (x2, y2), (x3, y3) (1, 3) (1.5, 6) (2.5, 8) x 1.67, y 5.67, sx .76, s y 2.52 r 11.67 35.67 1.51.67 65.67 2.51.67 85.67 (31)(.76)(2.52) .9538 Properties of Correlation r is a measure of the strength of the linear relationship between x and y. No units [like demand elasticity in economics (-infinity, 0)] -1 < r < 1 Values of r and scatterplots r near +1 r near -1 y r near 0 r near 0 y x x Properties (cont.) r has no unit Changing the units of variables does not change the correlation coefficient "r", because we get rid of all our units when we standardize (get z-scores). r = -0.75 z-score plot is the same for both plots r = -0.75 Properties (cont.) r ranges from -1 to+1 "r" quantifies the strength and direction of a linear relationship between 2 quantitative variables. Strength: how closely the points follow a straight line. Direction: is positive when individuals with higher X values tend to have higher values of Y. Properties of Correlation (cont.) r = -1 only if y = a + bx with slope b<0 r = +1 only if y = a + bx with slope b>0 10 20 y = 11 - x 8 y = 1 + 2x r=1 r = -1 15 6 Y 10 y 4 5 2 0 0 0 2 4 6 x 8 10 0 2 4 6 X 8 10 Properties (cont.) High correlation does not imply cause and effect CARROTS: Hidden terror in the produce department at your neighborhood grocery Everyone who ate carrots in 1920, if they are still alive, has severely wrinkled skin!!! Everyone who ate carrots in 1865 is now dead!!! 45 of 50 17 yr olds arrested in Raleigh for juvenile delinquency had eaten carrots in the 2 weeks prior to their arrest !!! Properties (cont.) Cause and Effect There is a strong positive correlation between the monetary damage caused by structural fires and the number of firemen present at the fire. (More firemen-more damage) Improper training? Will no firemen present result in the least amount of damage? Properties (cont.) Cause and Effect (1,2) (24,75) (1,0) (18,59) (9,9) (3,7) (5,35) (20,46) (1,0) (3,2) (22,57) x = fouls committed by player; y = points scored by same player r measures the strength of the linear relationship between x and y; it does not indicate cause and effect The correlation is due to a third “lurking” variable – playing time (x, y) = (fouls, points) correlation r = .935 Points 80 70 60 50 40 30 20 10 0 0 5 10 15 Fouls 20 25 30 Properties (cont.) r does not distinguish x & y The correlation coefficient, r, treats x and y symmetrically. r = -0.75 r = -0.75 "Time to swim" is the explanatory variable here, and belongs on the x axis. However, in either plot r is the same (r=-0.75). Outliers Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers. Just moving one point away from the general trend here decreases the correlation from -0.91 to -0.75 PROPERTIES (Summary) r is a measure of the strength of the linear relationship between x and y. No units [like demand elasticity in economics (-infinity, 0)] -1 < r < 1 r = -1 only if y = a + bx with slope b<0 r = +1 only if y = a + bx with slope b>0 correlation does not imply causation r does not distinguish between x and y r can be affected by outliers Correlation: Fuel Consumption vs Car Weight FUEL CONSUMP. (gal/100 miles) FUEL CONSUMPTION vs CAR WEIGHT r = .9766 7 6 5 4 3 2 1.5 2.5 3.5 WEIGHT (1000 lbs) 4.5 SAT Score vs Proportion of Seniors Taking SAT 88-89 SAT vs % Seniors Taking SAT r = -.868 88-89 SAT State Avg. IW ND 1075 1025 975 88-89 SAT 925 875 SC 825 0 20 40 DC NC 60 % Seniors that Took SAT 80 Standardization: Allows us to compare correlations between data sets where variables are measured in different units or when variables are different. For instance, we might want to compare the correlation between [swim time and pulse], with the correlation between [swim time and breathing rate]. When variability in one or both variables decreases, the correlation coefficient gets stronger ( closer to +1 or -1). Correlation only describes linear relationships No matter how strong the association, r does not describe curved relationships. Note: You can sometimes transform a non-linear association to a linear form, for instance by taking the logarithm. You can then calculate a correlation using the transformed data. Review examples 1) What is the explanatory variable? Describe the form, direction and strength of the relationship? Estimate r. r = 0.94 (in 1000’s) 2) If women always marry men 2 years older than themselves, what is the correlation of the ages between husband and wife? r=1 ageman = agewoman + 2 equation for a straight line Thought quiz on correlation 1. Why is there no distinction between explanatory and response variable in correlation? 2. Why do both variables have to be quantitative? 3. How does changing the units of one variable affect a correlation? 4. What is the effect of outliers on correlations? 5. Why doesn’t a tight fit to a horizontal line imply a strong correlation?