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1.2StatisticsCore

Measuring Correlation

The product moment correlation coefficient (PMCC), r, measures the strength and direction of a linear relationship between two variables. It always lies between −1 and 1.

24 min Regression, Correlation and Hypothesis Testing
Edexcel A Level Maths: 1.2 Measuring CorrelationCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Interpret the value of r
  • Know the range −1 ≤ r ≤ 1
  • Distinguish strength and direction
  • Read r from a calculator
1

What r tells you

The PMCC rr satisfies −1≤r≤1-1 \le r \le 1. Values near ±1\pm 1 indicate strong linear correlation; near 00 indicates weak or no linear correlation. The sign gives the direction.

−1≤r≤1-1 \le r \le 1
r measures linear correlation.
Example 1Interpret r=−0.92r = -0.92.
1Close to −1-1.
2Strong negative linear correlation.
Answer
Strong negative correlation.

Tip — r measures LINEAR correlation only — a strong curved pattern can still give small r.

2

Strength and direction

Sign: r>0r > 0 positive (both rise together), r<0r < 0 negative (one rises as the other falls). Magnitude: closer to 1 means stronger.

Formula recap

−1≤r≤1-1 \le r \le 1
Range of the PMCC.
r≈±1⇒strongr \approx \pm 1 \Rightarrow \text{strong}
Strength.

Common mistakes to avoid

Claiming r near 0 means no relationship at all.
It means no LINEAR relationship; a curved one may exist.
Saying correlation proves causation.
Correlation does not imply causation.

Key takeaways

  • PMCC r lies in [−1, 1].
  • Sign = direction; magnitude = strength.
  • r measures linear correlation only.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The PMCC always lies between:

    • −1-1 and 11
    • 00 and 11
    • −∞-\infty and ∞\infty
    • 00 and 100100
    Show answer and explanation

    Answer: −1-1 and 11

    By definition.

  2. r=−0.92r = -0.92 indicates:

    • strong negative correlation
    • weak negative
    • strong positive
    • no correlation
    Show answer and explanation

    Answer: strong negative correlation

    Near −1.

  3. r close to 0 means:

    • weak linear correlation
    • strong correlation
    • perfect correlation
    • causation
    Show answer and explanation

    Answer: weak linear correlation

    Little linear trend.

  4. The sign of r gives the:

    • direction
    • strength
    • sample size
    • units
    Show answer and explanation

    Answer: direction

    + or −.

  5. Correlation does NOT imply:

    • causation
    • a linear trend
    • a value of r
    • direction
    Show answer and explanation

    Answer: causation

    Key warning.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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