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?

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

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.

X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o positive r

X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o negative r

X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o zero r

X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o Non-linear relationship

Correlation Coefficients X dataY dataCoefficient interval/ratiointerval/ratioPearson r ordinalordinalSpearman rho dichotomousinterval/ratioPoint Biserial dichotomous dichotomousPhi dichotomous: having only two values

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

More on Dichotomous Variables If the correlation between gender and GPA is positive, it could mean that – females have higher GPAs, if males were 1’s and females were 2’s – males have higher GPAs, if females were 1’s and males were 2’s

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

What is the Coefficient of Determination? Measure of proportion of explained variance in Y based on X. Square r to get r 2. Example: r =.36r 2 =.13 We can explain 13% of the differences in Exam 2 scores by knowing Exam 1 scores.

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

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.

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.

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.