Karl Pearson's Coefficient of Correlation: Formula and Method

Understand Karl Pearson's Coefficient of Correlation

Key formulas to remember

Definition of r

r = Cov(X, Y) ÷ (σx × σy)

Cov(X, Y) = Σ(X − X̄)(Y − Ȳ) ÷ n = ΣXY ÷ n − X̄Ȳ.

Deviation-from-mean form

r = Σ(X − X̄)(Y − Ȳ) ÷ √[Σ(X − X̄)² × Σ(Y − Ȳ)²]

Use when the means are whole numbers. The factor n cancels out.

Direct method

r = [nΣXY − ΣXΣY] ÷ √{[nΣX² − (ΣX)²][nΣY² − (ΣY)²]}

Best when the values are small. n is the number of pairs.

Assumed mean method

r = [nΣdxdy − ΣdxΣdy] ÷ √{[nΣdx² − (Σdx)²][nΣdy² − (Σdy)²]}

dx = X − A and dy = Y − B, where A and B are assumed means. The same formula as direct, with d replacing X and Y.

Step-deviation method

u = (X − A) ÷ h, v = (Y − B) ÷ k; r = [nΣuv − ΣuΣv] ÷ √{[nΣu² − (Σu)²][nΣv² − (Σv)²]}

If h and k are both positive, r is the same as for X and Y. If exactly one of h or k is negative, r changes sign.

Limits of r

−1 ≤ r ≤ +1

r = ±1 means perfect linear correlation. r = 0 means no linear correlation.

Change of origin and scale

If u = (X − a) ÷ h and v = (Y − b) ÷ k with h, k > 0, then r(u, v) = r(X, Y)

r does not depend on origin or on positive scale.

How to solve Karl Pearson's Coefficient of Correlation questions

1. 1Count the pairs of observations and write n.
2. 2If the values are large, pick an assumed mean for each variable (near the middle) and, if all values are multiples of a common number, a step size. Convert to dx, dy or u, v.
3. 3Make columns for X, Y, XY, X² and Y² (or the same for the deviations). Fill every row.
4. 4Add each column and write the totals: ΣX, ΣY, ΣXY, ΣX², ΣY².
5. 5Calculate the numerator nΣXY − ΣXΣY, and the two terms under the root, nΣX² − (ΣX)² and nΣY² − (ΣY)².
6. 6Divide the numerator by the square root of the product of the two terms. Check that the sign matches the numerator.
7. 7Check that r lies between −1 and +1, then state the meaning: strong or weak, positive or negative.

Quickest way: Shift and shrink, then use the sums formula

Common mistakes in Karl Pearson's Coefficient of Correlation

• Writing (ΣX)² as ΣX² (or the reverse).

  The two look alike, but one is the square of the total and the other is the total of the squares.

  Fix: Keep separate columns. ΣX² comes from squaring each X and then adding. (ΣX)² comes from adding first and then squaring.

• Forgetting to multiply ΣXY by n in the numerator.

  Students remember the covariance form ΣXY ÷ n and mix it with the sums form.

  Fix: In the sums formula, every first term has n: nΣXY, nΣX² and nΣY². Write the formula before you start.

• Getting r outside the range −1 to +1 and still marking it.

  A calculation slip is not checked at the end.

  Fix: If |r| is more than 1, there is an error. Recheck the sums and the squares.

• Thinking a change of origin or scale changes r.

  Students confuse r with covariance or standard deviation, which do change with scale.

  Fix: Remember that r has no unit. Positive shifts and divisions leave r unchanged. Only a negative divisor on one variable flips the sign.

• Concluding that r = 0 means the variables are unrelated.

  The word 'no correlation' sounds like 'no relationship'.

  Fix: r = 0 means no linear relationship. A non-linear relationship can still exist.

• Using the assumed mean method with mistakes in the signs of deviations.

  Values below the assumed mean give negative deviations that students write as positive.

  Fix: Always compute X − A with its sign. Check that Σdx matches the gap between the actual mean and A.

Worked examples