Correlation

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Correlation

• We can often see the strength of the relationship
  between two quantitative variables in a
  scatterplot, but be careful. The two figures here
  are both of the same data, on different scales.
  The second seems to be a stronger association

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• Heres the formula (p.102) for Pearsons
  correlation coefficient
• This formula is not for computing r but for
  understanding r. Notice that the first step in
  this formula involves standardizing each x and y
  value and then multiplying the two standardized
  values (how many s.d.s above or below the means
  the xs and ys are...) together.
• When two variables x and y are positively
  associated their standardized values tend to be
  both positive or both negative (think of height
  and weight) so the product is positive.
• When two variables are negatively associated then
  if x for example is above the mean, the y tends
  to be below the mean (and vice versa) so the
  product is negative.

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• The correlation coefficient, r, is a numerical
  measure of the strength of the linear
  relationship between two quantitative variables.
• It is always a number between -1 and 1.
  Positive r positive association
• Negative r negative association
• r1 implies a perfect positive relationship
  points falling exactly on a straight line with
  positive slope
• r-1 implies a perfect negative relationship
  points falling exactly on a straight line with
  negative slope
• r0 implies a very weak linear relationship

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• Correlation makes no distinction between
  explanatory response variables doesnt matter
  which is which
• Both variables must be quantitative
• r uses standardized values of the observations,
  so changing scales of one or the other or both of
  the variables doesnt affect the value of r.
• r measures the strength of the linear
  relationship between the two variables. It does
  not measure the strength of non-linear or
  curvilinear relationships, no matter how strong
  the relationship is
• r is not resistant to outliers be careful about
  using r in the presence of outliers on either
  variable

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• To explore how extreme outlying observations
  influence r, see the applet on Correlation and
  Regression in the ebook...
• Homework
• Read section 2.2
• Using technology to draw the scatterplots and do
  the computations, work problems 2.29 - 2.32,
  2.35, 2.39, 2.41, 2.43 2.46 (applet), 2.50,
  2.51