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Expected values, covariance, correlation and expected values

Expected values, covariance, correlation and expected values. Introduction to Bivariate Regression . Review . Mean Mode Median Freq Variance Standard deviation.

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Expected values, covariance, correlation and expected values

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  1. Expected values, covariance, correlation and expected values Introduction to Bivariate Regression

  2. Review • Mean • Mode • Median • Freq • Variance • Standard deviation

  3. Is the perception that the majority of Russians believe the same way you do related to how often you discuss politics with friends?

  4. Is this a causal relationship? Majority of Russians believe the same Discussions of politics with friends Discussions of politics with friends Majority of Russians believe the same

  5. When it comes to politics, how close do you think your opinions are to the opinions of the majority of Russians? very close, rather close, not very close, not close at all freq vars = majrcl / stats = mean stddev var.

  6. How often do you do the following discuss political questions with friends, neighbors, or coworkers almost never, a few times a year, a few times a month, a few times a week, or practically every day? freq vars = discfrnd / stats = mean stddev var.

  7. Review standard deviation and variance • Variance: for each unit or observation, it is the distance from the mean squared and then divide by the number of units • Standard deviation – squareroot of variance • since variance is in squared units, it doesn’t make any sense. The standard deviation can be understood in terms of the original measurement unit

  8. Calculating variance and standard deviations

  9. Review: Units, mean, variance and standard deviation majrcl discfrnd 2.00 4.00 2.00 3.00 . 4.00 . 1.00 2.00 1.00 3.00 4.00 3.00 3.00 3.00 3.00 . 3.00 2.00 2.00 . 3.00 3.00 3.00 3.00 4.00 3.00 5.00 3.00 1.00 3.00 3.00

  10. Expected value v. probability • If our population set of numbers is: 1,1,3,3,17, then the expected value is 5, even though P(5) = 0. • Suppose we know that E(X) = 5 with the equation y = 5 + 7x. • What is E(Y)?

  11. Expected values What is the expected value of majrcl? What is the range? Mode? Why are there 63 missing? What is the expected value of discfrnd? Why is the standard deviation and variance so high?

  12. Crosstab

  13. Causation • Time ordering • Covariation

  14. Co-variation from variation? • (xi - xmean)^2/n average distance between the mean of x and each x value, squared • aka (xi - xmean) (xi - xmean)/n

  15. Covariation? (xi - xmean) * (yi - ymean) / n-1

  16. Covariation • covariance can take any value • negative infinity to positive infinity

  17. Intuitive explanation (xi - xmean) * (yi - ymean) / n-1 • When x and y are high at the same time and x and y are low at the same time, then the covariance is positive • They are both higher than their means and so the products being added together are positive

  18. Plot showing positive covariance Mean urban % Mean female literacy

  19. Intuitive explanation (xi - xmean) * (yi - ymean) / n-1 • When x is low when y is high and vice versa, then the covariance is negative • They are both higher than their means and so the products being added together are negative

  20. Plot showing negative covariance Mean calorie intake Mean infant mortality

  21. Intuitive explanation (xi - xmean) * (yi - ymean) / n • When sometimes: • x and y are high at the same time and x and y are low at the same time • And about half of the other time • x is low when y is high and vice versa • Then the covariance is about 0 • High positive numbers are added to high negative numbers

  22. Plot showing no covariance Mean GDP Mean crop production

  23. Covariance is a function of… • Variance (standard deviation) of x • Variance (standard deviation) of y • Relationship between x and y

  24. How can you compare a covariance of 132 and 134,847? • 134, 847 could be high variance of x, high variance of y, high variance of both variables, or a high relationship between x and y? • Not that helpful?

  25. How can you change the covariance to a number that tells you only the magnitude of the relationship between x and y? • Divide by the standard deviation of x * the standard deviation of y • Correlation = (x-xmean)*(y-ymean) /Sd(x) * sd (y) • Pearson r ranges from -1 to +1 • Weak correlation = .1 • moderate correlation = .4 • strong correlation = .7

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