Week 1 · Correlation: direct and inverse relations between two variables, scatter diagrams and the scale of the correlation coefficient r from −1 to 1 · 3 lessons
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The correlation coefficient r: the scale from −1 to 1, perfect, strong, weak and no correlation, and comparing coefficients by size
Statistics Sec 3 Term 1, Unit 1, Lesson 1
GoalRead the correlation coefficient r on its scale from −1 to 1, and compare coefficients by their size without the sign.
New words
correlation coefficient r · a number from −1 to 1 that measures a straight-line relationperfect correlation · r = 1 or r = −1: all dots on one straight linestrength · the size of r without its sign, |r|
Explanation
Looking at a cloud of dots is not enough when two people see it differently, so we measure. The correlation coefficient r is a single number with −1 ≤ r ≤ 1. Its sign gives the type: positive for a direct correlation, negative for an inverse one. Its size gives the strength. If r = 1, all the dots lie on one rising straight line (perfect direct correlation); if r = −1, on one falling line (perfect inverse correlation); r = 0 means no straight-line correlation. The nearer r is to 1 or to −1, the stronger the correlation; the nearer to 0, the weaker. To compare two coefficients compare their sizes without signs: r = −0.9 is stronger than r = 0.7, because 0.9 > 0.7. A value such as 1.3 or −1.2 is impossible and shows an arithmetic slip. Finally, r measures only straight-line relations: the area of a square is exactly its side times itself, yet for sides 1 to 6 the dots lie on a curve and r is about 0.98, not 1. Next week we shall see how r is calculated.
Worked examples
The three data sets of the last lesson have r ≈ 0.97 for A (revision and mark), r ≈ −0.87 for B (altitude and temperature) and r ≈ −0.04 for C (seat number and mark), found with the formula. A is a strong direct correlation, B a strong inverse one (its size 0.87 is less than A’s 0.97, so B is a little weaker), and C is practically none.
Which is the strongest of 0.62, −0.78, 0.15, −0.31? Compare sizes: 0.62, 0.78, 0.15, 0.31. The largest is 0.78, so −0.78 is the strongest (and it is inverse). Which is the strongest inverse among 0.2, −0.45, −0.9, 0.95? Inverse means negative, so only −0.45 and −0.9 qualify: the answer is −0.9, although 0.95 is stronger overall.
A square of side x has area x². For x = 1, 2, …, 6 the areas are 1, 4, 9, 16, 25, 36. The relation is exact and increasing, but the dots lie on a curve, so the straight-line measure gives r ≈ 0.98 and not 1.
Class activity
Cut seven strips of paper labelled −1, −0.6, −0.2, 0, 0.3, 0.7, 1. Put them in order on a table like the scale, then pick two and say which one shows the stronger correlation and which one is direct or inverse.
Practice
1
Which shows the stronger correlation: r = 0.9 or r = −0.95?
r = −0.95, because 0.95 > 0.9.
2
A student reports r = 1.2 for two variables. What do you say?
It is impossible, since r is always between −1 and 1: he made a mistake in the calculation.
3
The dots of a diagram lie almost exactly on a falling line. Which r is likeliest: −0.97, −0.1 or 0.5?
−0.97: a strong inverse correlation.
4
Ahmed finds r = 0.8 for hours of revision and marks. Mona finds r = −0.8 for altitude and temperature. Do they have the same strength? The same type?
The same strength (both sizes are 0.8) but different types: Ahmed’s is direct, Mona’s is inverse.
5
A pupil says that r = 0.6 means 60% of the dots lie on the line. Is he right?
No. r is not a percentage of dots. It measures how closely the dots follow a straight line; with r = 0.6 perhaps no dot lies exactly on it.
