CORRELATION COEFFICIENTS

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CORRELATION COEFFICIENTS

• What Does a Correlation Coefficient Indicate?
• What is a Scatterplot?
• Correlation Coefficients
• What Could a Low r mean?
• What is the Coefficient of Determination?
• How Do We Test r for Significance?

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What Does a Correlation Coefficient Indicate?

• Strength of relationship
• 0 means no relationship at all
• -1 or 1 means perfectly related
• Direction of relationship
• positive variable X increases as variable Y
  increases
• negative variable X decreases as variable Y
  increases

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What is a Scatterplot?

• A correlation is graphically displayed using a
  scatterplot.
• The scatterplot is a graph with one variable on
  each axis, and a mark for each individual
  measured.
• Scatterplots are useful for estimating the
  direction, strength, and nature of the
  relationship.

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high
positive r
o
o
o
o
o
o
o
Y
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
low
high
X
5
high
o
o
o
negative r
o
o
o
Y
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
low
high
X
6
high
o
o
o
o
zero r
o
o
o
o
o
o
Y
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
low
high
X
7
Non-linear relationship
high
o
o
o
o
o
o
o
o
o
o
o
o
Y
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
low
high
X
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Correlation Coefficients

• X data Y data Coefficient
• interval/ratio interval/ratio Pearson r
• ordinal ordinal Spearman rho
• dichotomous interval/ratio Point Biserial
• dichotomous dichotomous Phi
• dichotomous having only two values

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More on Dichotomous Variables

• With dichotomous variables, whether r is
  negative or positive depends on how the numbers
  were assigned

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More on Dichotomous Variables

• If the correlation between gender and GPA is
  positive, it could mean that
• females have higher GPAs, if males were 1s and
  females were 2s
• males have higher GPAs, if females were 1s and
  males were 2s

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What Could a Low r Mean?

• Lack of a relationship.
• Unreliable measurement.
• Non-linear relationship.
• Restricted range full range of scores not
  measured on both variables

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What is the Coefficient of Determination?

• Measure of proportion of explained variance in Y
  based on X.
• Square r to get r2.
• Example r .36 r2 .13
• We can explain 13 of the differences in Exam
  2 scores by knowing Exam 1 scores.

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How Do We Test r for Significance?

• Any test of significance requires assumptions
  for the test to be valid.
• For a correlation coefficient, the assumptions
  are
• appropriate measurement scales for correlation
  coefficient
• independent pairs of scores
• normally distributed variables
• if a relationship exists, it is linear

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How Do We Test r for Significance?

• Decide whether to do a one-tailed or two-tailed
  test.
• One-tailed directional hypothesis (positive r or
  negative r)
• Two tailed nondirectional hypothesis
• It is always safe to do a two-tailed test.

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How Do We Test r for Significance?

• Determine the alpha level. The conventional
  level is .05.
• Determine the degrees of freedom (df), which
  affects the shape of the sample distribution. For
  r, the df N-2.

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How Do We Test r for Significance?

• When computing by hand, find the critical value
  of r in an r table. If the calculated r beats the
  critical value, the result is significant.
• When using computer software, look for the
  significance level of r. If the significance
  level is smaller than your alpha level, the
  result is significant.