Pearson's correlation

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Pearsons correlation
Diane S. Mendoza
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• It is named after Karl Pearson who developed the
  correlational method to do agricultural research.
  
• designated by the Greek letter rho ()
• The product moment part of the name comes from
  the way in which it is calculated by summing up
  the products of the deviations of the scores from
  the mean.
• A correlation is a number between -1 and 1 that
  measures the degree of association between two
  variables (call them X and Y).
• A positive value for the correlation implies a
  positive association
• A negative value for the correlation implies a
  negative or inverse association

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The formula for the Pearson correlation
Suppose we have two variables X and Y with means
XBAR and YBAR respectively and standard
deviations SX and SY respectively. The
correlation is computed as
as the sum of the product of the Z-scores for the
two variables divided by the number of scores.
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If we substitute the formulas for the Z-scores
into this formula we get the following formula
for the Pearson Product Moment Correlation
Coefficient which we will use as a definitional
formula.
The numerator of this formula says that we sum up
the products of the deviations of a subjects X
score from the mean of the Xs and the deviation
of the subjects Y score from the mean of the Ys.
This summation of the product of the deviation
scores is divided by the number of subjects times
the standard deviation of the X variable times
the standard deviation of the Y variable
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• When will a correlation be positive
• Suppose that an X value was above average and
  that the associated Y value was also above
  average. Then the product would be the product of
  two positive numbers which would be positive.
• If the X value and the Y value were both below
  average then the product above would be of two
  negative numbers which would also be positive.
• Therefore a positive correlation is evidence of
  a general tendency that large values of X are
  associated with large values of Y and small
  values of X are associated with small values of Y.

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• When will a correlation be negative
• Suppose that an X value was above average and
  that the associated Y value was instead below
  average. Then the product would be the product of
  a positive and a negative number which would make
  the product negative.
• If the X value was below average and the Y value
  was above average then the product above would
  be also be negative.
• Therefore a negative correlation is evidence of
  a general tendency that large values of X are
  associated with small values of Y and small
  values of X are associated with large values of Y.

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Interpretation of the correlation
coefficient The correlation coefficient measures
the strength of a linear relationship between two
variables. The correlation coefficient is always
between -1 and 1. The closer the correlation is
to /-1 the closer to a perfect linear
relationship. Here is to interpret
correlations. -1.0 to -0.7 strong negative
association. -0.7 to -0.3 weak negative
association. -0.3 to 0.3 little or no
association. 0.3 to 0.7 weak positive
association. 0.7 to 1.0 strong positive
association.
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• Lets calculate the correlation between Reading
  (X) and Spelling (Y) for the 10 students. There
  is a fair amount of calculation required as you
  can see from the table below. First we have to
  sum up the X values (55) and then divide this
  number by the number of subjects (10) to find the
  mean for the X values (5.5). Then we have to do
  the same thing with the Y values to find their
  mean (10.3).

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Formula
We then calculate
The correlation we obtained was -.36 showing us
that there is a small negative correlation
between reading and spelling. The correlation
coefficient is a number that can range from -1
(perfect negative correlation) through 0 (no
correlation) to 1 (perfect positive correlation).
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The computational formula for the Pearsonian r is

• By looking at the formula we can see that we need
  the following items to calculate r using the raw
  score formula
• The number of subjects N
• The sum of each subjects X score times the Y
  score summation XY
• The sum of the X scores summation X
• The sum of the Y scores summation Y
• The sum of the squared X scores summation X
  squared
• The sum of the squared Y scores summation Y
  squared

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In we plug each of these sums into the raw score
formula we can calculate the correlation
coefficient
We can see that we got the same answer for the
correlation coefficient (-.36) with the raw score
formula as we did with the definitional formula.
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GRACIAS!