Correlation

Summary — What This Chapter Covers

• PMCC (r) — measures linear correlation between two continuous, normally distributed variables.
• Spearman's Rank (rₛ) — a rank-based alternative used when data isn't linear, isn't continuous, or isn't normally distributed.
• Coding data doesn't change correlation — as long as the coding itself is linear (e.g. subtracting a constant, dividing by a constant).
• Hypothesis testing for correlation — using your sample's r or rₛ to decide whether there's genuinely a correlation in the whole population, or whether H₀ (no correlation) should be accepted.
• Always interpret your final answer in context — never just leave a number floating with no meaning attached.

1. Product Moment Correlation Coefficient (PMCC)

Think of PMCC as answering one question: "If I drew a straight line through this scatter of points, how well would that line actually fit?" It only cares about straight-line (linear) relationships, and it only works properly when both variables are continuous and roughly normally distributed — things like height, temperature, or exam marks, rather than categories or rankings.

The result, r, always lands somewhere between −1 and 1:

The Formula

Worked Example — Physics vs. Chemistry Marks

2. Spearman's Rank Correlation Coefficient