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The Correlation Coefficient. Social Security Numbers. A Scatter Diagram. The Point of Averages. Where is the center of the cloud? Take the average of the x -values and the average of the y -values; this is the point of averages . It locates the center of the cloud.

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the point of averages
The Point of Averages
  • Where is the center of the cloud?
  • Take the average of the x-values and the average of the y-values; this is the point of averages.
  • It locates the center of the cloud.
  • Similarly, take the SD of the x-values and the SD of the y-values.
the correlation coefficient6
The Correlation Coefficient
  • An association can be stronger or weaker.
  • Remember: a strong association means that knowing one variable helps to predict the other variable to a large extend.
  • The correlation coefficient is a numerical value expressing the strength of the association.
the correlation coefficient7
The Correlation Coefficient
  • We denote the correlation coefficient by r.
  • If r = 0, the cloud is completely formless; there is no correlation between the variables.
  • If r = 1, all the points lie exactly on a line (not necessarily x = y) and there is perfect correlation.
the correlation coefficient9
The Correlation Coefficient
  • What about negative values?
  • The correlation coefficient is between –1 and 1, negative shows negative association, positive indicates positive association.
  • Note that –0.90 shows the same degree of association as +0.90, only negative instead of positive.
computing the correlation coefficient
Computing the Correlation Coefficient
  • Convert each variable to standard units.
  • The average of the products gives the correlation coefficient r.

r = average of

(x in standard units)  (y in standard units)

example
Example

We mustfirstconvert to standard units.

Find the average and the SD of the x-values: average = 4, SD = 2.

Find the deviation: subtract the average from each value, and divide by the SD.

Then do the same for the y-values.

example14
Example
  • Finally, take the average of the products
  • In this example, r = 0.40.

r = average of

(x in standard units)  (y in standard units)

the sd line
The SD line
  • If there is some association, the points in the scatter diagram cluster around a line. But around which line?
  • Generally, this is the SDline. It is the line through the point of averages.
  • It climbs at the rate of one vertical SD for each horizontal SD.
  • Its slope is (SD of y) / (SD of x) in case of a positive correlation, and –(SD of y) / (SD of x) in case of a negative correlation.
five point summary
Five-point Summary
  • Remember the five-point summary of a data set: minimum, lower quartile, median, upper quartile, and maximum.
  • A five-point summary for a scatter plot is: average x-values, SD x-values, average y-values, SD y-values, and correlation coefficient r.