Correlation (I)

Predicting correlation between variables (practice vs performance)

• Relationship: Positive association results from:

• Variables: Practice ↔ Performance

• Meaning: As practice increases, performance improves

• Direction: Bidirectional link (they influence each other)

• Key idea: More maths practice → better maths performance

Do changes in one variable tend to be associated with specific changes in the other variable?

• Variables measured:

  • Practice (hours/week), mean = 4.6

  • Performance (problems solved), mean = 6.2

• Research question: Do the variables covary?

• Key idea: Looking for association, not just averages

• Method implication: Need to compare patterns across individuals (not means alone)

• Conclusion: Cannot tell from means alone whether practice and performance are related; need correlation/relationship analysis

How do we assess the relationship between two variables?

• Goal: Determine if two variables are related (e.g., practice & performance)

• Pictorial method: Scatterplot

  • Shows pattern of association visually

• Numerical method: Correlation coefficient

  • Quantifies strength and direction of relationship

• Key idea: Use both visual and statistical approaches for full understanding

How do you create a scatterplot in IBM SPSS?

• Go to Graphs → Legacy Dialogs → Scatter/Dot

• Select Simple Scatter → Define

• Assign variables:

  • Drag first variable to X-axis (IV in experiments)

  • Drag second variable to Y-axis (DV in experiments)

  • For correlations, either variable can go on either axis

• Click OK to produce scatterplot

What types of relationships can be seen in a scatterplot?

• Positive association: Higher practice → higher performance

• Negative association: Higher practice → lower performance

• Perfect positive association: Exact match; increases in practice perfectly predict increases in performance

• Perfect negative association: Exact match; increases in practice perfectly predict decreases in performance

• No association: No consistent pattern between practice and performance

• Non-linear association: Relationship exists but follows a curved (not straight-line) pattern

What three aspects of a relationship can a scatterplot show?

• Direction:

  • Upward trend (bottom-left → top-right) = positive

  • Downward trend (top-left → bottom-right) = negative

• Strength:

  • Points tightly clustered around a line = strong relationship

  • More scattered points = weaker relationship

• Form:

  • Straight line fit = linear relationship

  • Curve fits better = non-linear relationship

How do we interpret a scatterplot and quantify the relationship?

• Direction: Upward line (bottom-left → top-right) = positive relationship

• Strength: Points closely clustered around the line = strong relationship

• Form: Straight-line fit = linear relationship

• Numerical assessment: Use Pearson’s correlation coefficient (r)

• Important condition: Pearson’s r is only appropriate for interval-scale data

How do you calculate Pearson’s r using Z-scores?

• 1. Convert all raw scores to Z-scores

• 2. Multiply paired Z-scores for each participant

• 3. Sum all products of Z-score pairs

  • 3. (a) Divide by (number of cases − 1)

  • Example: (0.66 + 0.59 + 0.48 + 1.82 + 0.14) ÷ 4 = 0.92

What are key steps and rules when calculating Pearson’s r using Z-scores?

• Z-score formula: (Raw score − Mean) ÷ SD

• Purpose of Z-scores: Standardises data (Mean = 0, SD = 1) to show distance from the mean

• Interpretation: Larger distance from mean → larger impact on correlation

• Multiplying Z-scores:

  • Both positive or both negative → positive product

  • One positive + one negative → negative product

• Key idea: Pearson’s r is based on the pattern of these Z-score products across participants

What is Pearson’s r formula and how do we interpret it?

• Range: -1 to +1

  • +1 = perfect positive correlation

  • -1 = perfect negative correlation

  • 0 = no linear correlation

• Positive r: Most Z-score products are positive (both scores same sign)

• Negative r: Most products are negative (opposite signs)

• Strength idea: Farther from mean → larger product → stronger r

• Note: Different formulas exist, but all give the same Pearson’s r result

When does a perfect positive correlation occur?

• Occurs when Pearson’s r = +1

• Each participant has equal Z-scores for both variables (Z_X = Z_Y)

• Both variables deviate from their means by the same amount and in the same direction

• Pattern: as one variable increases, the other increases perfectly in step

• Key idea: identical standardized deviations → perfect positive association

When does a perfect negative correlation occur?

• Occurs when Pearson’s r = −1

• Each participant has equal Z-scores in magnitude but opposite signs (Z_X = − Z_Y)

• Both variables are equally distant from their means but in opposite directions

• Pattern: as one variable increases, the other decreases perfectly in step

• Key idea: mirror-image standardized deviations → perfect negative association

When is a correlation undefined in a scatterplot?

• Occurs when one variable has no variation (all scores identical)

• Scatterplot becomes:

  • Horizontal line (if Y is constant)

  • Vertical line (if X is constant)

• No relationship can be calculated because covariance is zero/undefined in this case

• Result: Pearson’s r cannot be computed

• In SPSS: output shows correlation is undefined or not computable

How do you obtain Pearson’s r in IBM SPSS?

• Go to Analyze → Correlate → Bivariate

• Move variables into the Variables box

• Select Pearson under Correlation Coefficients

• Choose one-tailed or two-tailed test of significance

• (Optional) Click Options to request:

  • Means

  • Standard deviations

  • Cross-product deviations

  • Covariance

• Click OK to run the analysis

How to understand the IBM-SPSS output for Pearson’s r?

How do you report results of a Pearson correlation?

• State that a Pearson correlation was conducted

• Describe the relationship (e.g., positive/negative association)

• Report whether it is statistically significant

• Include key details:

  • r value (strength/direction)

  • sample size (n)

  • p-value (significance level)

  • one- or two-tailed test

• Example format: r = .93, n = 5, p < .05, two-tailed

Outline how to describe results of correlation?

A Pearson correlation was computed for the relationship between maths practice and performance. This revealed that these variables were, as predicted, positively related and that the correlation was statistically significant (r = .93, n = 5, p < .05, two-tailed).