Correlations

• Correlation


  A measure of both the strength and direction of the relationship between two variables
• Correlation


  A standardization of covariance
• Direction of correlation


  • Positive: two variables move (or vary) in the same direction
  • Negative: two variables move (or vary) in the opposite direction
  • Curvilinear: as one variable goes up, the other goes one direction but then changes directions
  • Zero: no relationship between two variables
• Strength of correlation


  Signal to noise ratio - the stronger the correlation, the less error/noise in the relationship
• Pearson's Product Moment Correlation Coefficient (r)

  A statistic that examines the linear relationship between two continuous variables. It measures the direction & strength of the relationship.
• Pearson's Correlation Coefficient (r)


  • Range: -1 ≤ r ≤ 1
  • Positive Correlation: 0 < r ≤ 1
  • Negative Correlation: -1 ≤ r < 0
  • No Correlation: r = 0
• Coefficient of Determination (r2)

  Percentage of variance in one variable that is accounted for by the other variable
• Relationship between r and r2

  The higher the absolute value of r, the stronger the relationship between the variables
• Interpreting strength of correlation (r)

  • .4 to 1.0 (-.4 to -1.0): Large effect (strong relationship)
  • .2 to .4 (-.2 to -0.4): Medium effect (moderate relationship)
  • .1 to .2 (-.1 to -.2): Small effect (small relationship)
  • .0 to .1 (.0 to -.1): Weak/No effect (weak relationship)
• Hypothesis testing for correlations

  1. Null hypothesis: 𝐻଴: 𝑟௫௬ = 0

  2. Alternative hypothesis: 𝐻ଵ: 𝑟௫௬ ≠ 0, 𝑟௫௬ > 0, 𝑟௫௬ < 0

  3. Test statistic: Ratio of 'thing we want' to 'thing we don't want'

  4. P-values: Probability of obtaining a test statistic this large or larger, assuming the null hypothesis is true
• Equation for Pearson's r


  𝑟௫௬ = 𝐶𝑜𝑣𝑎𝑟𝑖𝑒𝑛��𝑒 𝑜𝑓 𝑋 & 𝑌 / 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑒𝑛𝑐𝑒 𝑜𝑓 𝑋 & 𝑌
• The closer r is to 1 or -1, the more covariance between x and y, and the less error variance
• Reading and writing correlations in APA style


  • r(df) = .xx, p = .yy
  • df = N - 2, where N is the number of people in the analysis
• Writing results


  • There was a significant positive correlation between X and Y, r(65) = .21, p = .03.