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Exploring Relationships Between Variables

Exploring Relationships Between Variables. Scatterplots and C orrelation. Objectives. Scatterplots Scatterplots Explanatory and response variables Interpreting scatterplots Outliers Categorical variables in scatterplots. Correlation The correlation coefficient “r”

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Exploring Relationships Between Variables

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  1. Exploring Relationships Between Variables Scatterplots and Correlation

  2. Objectives Scatterplots • Scatterplots • Explanatory and response variables • Interpreting scatterplots • Outliers • Categorical variables in scatterplots Correlation • The correlation coefficient “r” • r does not distinguish x and y • r has no units • r ranges from -1 to +1 • Influential points

  3. Basic Terminology • Univariate data: 1 variable is measured on each sample unit or population unit 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)

  4. 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

  5. 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?

  6. Scatterplots • Useful method to graphically describe the relationship between 2 quantitative variables

  7. 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.

  8. Focus on Three Features of a Scatterplot Look for an overall pattern regarding … • Shape - ? Approximately linear, curved, up-and-down? • Direction - ? Positive, negative, none? • Strength - ? Are the points tightly clustered in the particular shape, or are they spread out? … and deviations from the overall pattern: Outliers 

  9. 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)

  10. Response (dependent) variable: blood alcohol content y x Explanatory (independent) variable: number of beers Explanatory and response variables A response variablemeasures or records an outcome of a study. An explanatory variableexplains 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.

  11. SAT Score vs Proportion of Seniors Taking SAT 2005 IW IL NC 74% 1010

  12. 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

  13. 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. Correlation can only be used to describe quantitative variables. Categorical variables don’t have means and standard deviations.

  14. Correlation: Fuel Consumption vs Car Weight r = .9766

  15. Example: calculating correlation • (x1, y1), (x2, y2), (x3, y3) • (1, 3) (1.5, 6) (2.5, 8) Automate calculation of the correlation! (Excel, statcrunch, calculator, etc.)

  16. 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

  17. 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.

  18. 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 y = 1 + 2x y = 11 - x

  19. 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 !!!

  20. 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?

  21. 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) • r measures the strength of the linearrelationship between x and y; it does not indicate cause and effect • correlation r = .935 x = fouls committed by player; y = points scored by same player The correlation is due to a third “lurking” variable – playing time

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