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Analyzing Variable Relationships through Scatterplots and Correlation Coefficients

Understand the relationship between variables through scatterplots and correlation coefficients. Learn to interpret strength and direction in bivariate data analysis. Determine correlations using Pearson's coefficient and explore properties of correlations. Discover how to calculate, interpret, and identify correlations effectively in this comprehensive guide.

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Analyzing Variable Relationships through Scatterplots and Correlation Coefficients

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  1. Chapter 5Summarizing Bivariate Data Correlation

  2. Variables: Response variable (y) measures an outcome (dependent) Explanatory variable (x) helps explain or influence changes in a response variable (independent)

  3. Suppose we found the age and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the age and weight of these adults?

  4. Does there seem to be a relationship between age and weight of these adults?

  5. Suppose we found the height and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the height and weight of these adults? Is it positive or negative? Weak or strong?

  6. Does there seem to be a relationship between height and weight of these adults?

  7. When describing relationships between two variables, you should address: • Direction (positive, negative, or neither) • Strength of the relationship (how much scattering?) • Form ( linear or some other pattern) • Unusual features (outliers or influential points) And ALWAYS in context of the problem!

  8. Identify as having a positive association, a negative association, or no association. + • Heights of mothers & heights of their adult daughters - • Age of a car in years and its current value + • Weight of a person and calories consumed • Height of a person and the person’s birth month NO • Number of hours spent in safety training and the number of accidents that occur -

  9. The closer the points in a scatterplot are to a straight line - the stronger the relationship. The farther away from a straight line – the weaker the relationship

  10. Correlation measures the direction and the strength of a linear relationship between 2 quantitative variables.

  11. Find the mean and standard deviation of the heights and weights of the 10 students:

  12. Find the mean and standard deviation of the heights and weights of the 10 students:

  13. Correlation Coefficient (r)- • A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data • Pearson’s sample correlation is used most • parameter - r (rho) • statistic - r

  14. Calculate r. Interpret r in context. r = .9964 There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

  15. Strong correlation No Correlation Moderate Correlation Weak correlation Properties of r(correlation coefficient) • legitimate values of r is [-1,1]

  16. Some Correlation Pictures

  17. Some Correlation Pictures

  18. Some Correlation Pictures

  19. Some Correlation Pictures

  20. Some Correlation Pictures

  21. Some Correlation Pictures

  22. x (in mm) 12 15 21 32 26 19 24 y 4 7 10 14 9 8 12 Find r. Interpret r in context. .9181 There is a strong, positive, linear relationship between speed limit and the number of weekly accidents.

  23. value of r is not changed by any transformations x (in mm) 12 15 21 32 26 19 24 y 4 7 10 14 9 8 12 Find r. Change to cm & find r. Do the following transformations & calculate r 1) 5(x + 14) 2) (y + 30) ÷ 4 .9181 .9181 The correlations are the same. STILL = .9181

  24. value of r does not depend on which of the two variables is labeled x Switch x & y & find r. Type: LinReg L2, L1 The correlations are the same.

  25. value of r is non-resistant x 12 15 21 32 26 19 24 y 4 7 10 14 9 8 22 Find r. Outliers affect the correlation coefficient

  26. value of r is a measure of the extent to which x & y are linearly related Find the correlation for these points: x -3 -1 1 3 5 7 9 Y 40 20 8 4 8 20 40 What does this correlation mean? Sketch the scatterplot r = 0, but has a definite relationship!

  27. Correlation makes no distinction between explanatory and response variable. It is unitless. • 2) Correlation does not change when we change the units of measurement of x, y, or both. • 3) Correlation requires both variables to be quantitative. • 4) Correlation does not describe curved relationship between variables, no matter how strong. Only the • linear relationship between variables. • 5) Like the mean and standard deviation, the correlation is not resistant: r is strongly affected by a few • outlying observations.

  28. Correlation does not imply causation Correlation does not imply causation Correlation does not imply causation

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