Correlation — Notes

Correlation is the degree and direction of the linear relationship between two variables. It is positive when the variables move in the same direction (both rise and fall together) and negative when they move in opposite directions. It is linear when the dots of a scatter diagram lie around a straight line and non-linear when they follow a curve.

Imagined examples: temperature and cold drink sales at a juice corner (positive); temperature and hot soup sales at a hill stall (negative); shoe size and marks of students (no correlation).

A scatter diagram plots one dot for each pair of values, with one variable on each axis. It shows the direction of the relation (rising or falling) and its closeness (how near the dots lie to a line).

Perfect positive correlation puts all dots on a rising line (coefficient 1); perfect negative on a falling line (−1); high correlation puts the dots close to a line; low correlation spreads them; no correlation shows no pattern (about 0). In the imagined sets of seven dots drawn here the coefficients are 0.86, −0.86, 0.25 and 0.

A high correlation does not prove that one variable causes the other. A third variable may affect both (in an imagined seaside town, hot weather raises both ice cream sales and sunburn: the coefficient for the imagined five months is 0.96), the link may be chance, or the direction of influence may be the reverse. A link caused by chance or by a third variable is a spurious correlation.

1. It lies between −1 and +1.
2. It has no unit.
3. It is not affected by a change of origin or scale: adding the same number to every value, or multiplying or dividing every value by the same positive number, leaves it unchanged. Multiplying one variable by a negative number reverses its sign.
4. A coefficient of 0 shows no linear relation; a curved relation may still exist (for the imagined temperature and electricity use of Himgiri the coefficient is 0 although the dots form a U).

For the imagined Lotus Valley data the coefficient is 0.8; after adding 2 to the hours it is 0.8, and after dividing the marks by 5 it is 0.8.

r = Σxy ÷ √(Σx² × Σy²), where x and y are the deviations of the two series from their means. Work in a table of deviations, products and squares.

For the imagined Lotus Valley students: Σxy = 40, Σx² = 10, Σy² = 250, so r = 40 ÷ 50 = 0.8.

The actual-value form is r = (nΣxy − ΣxΣy) ÷ √[(nΣx² − (Σx)²)(nΣy² − (Σy)²)]; for the same data it gives 200 ÷ √(50 × 1250) = 0.8. The sign of Σxy is the sign of r.

rs = 1 − 6ΣD² ÷ (n³ − n), where D is the difference between the two ranks of an item and n is the number of items. Rank 1 is given to the highest value. It is used for qualitative data that can only be ranked and when extreme values are present. The differences D always add up to zero.

Repeated ranks: tied values get the average of the ranks they cover, and for each group of m tied items (m³ − m) ÷ 12 is added to ΣD²: rs = 1 − 6[ΣD² + Σ(m³ − m) ÷ 12] ÷ (n³ − n). Imagined examples: two judges of a debate give rs = 0.8; six students with one tied pair give ΣD² = 6.5, correction 0.5 and rs = 0.8.

As a rough guide, a value near +1 shows a very strong positive relation (for example +0.9), a value of about +0.3 a weak positive one, a value near 0 no linear relation, and a value such as −0.7 a fairly strong negative one. These bands are not fixed rules, and no value alone proves cause.

Use Karl Pearson's coefficient for exactly measured values and rank correlation for ranks, qualitative data and data with extreme values. For five imagined shops, one with very large sales, Karl Pearson's coefficient is 0.75 and rank correlation is 1.