CA Foundation · Quantitative Aptitude

Correlation and Regression: formula sheet

Full chapter guide

Key formulas

Range of correlation coefficient: -1 ≤ r ≤ +1; r can never be more than +1 or less than -1. A value outside this range means a calculation error.
Perfect positive correlation: r = +1; All points lie exactly on a rising straight line.
Perfect negative correlation: r = -1; All points lie exactly on a falling straight line.
Zero (no linear) correlation: r = 0; No linear relationship. Non-linear dependence may still exist.
Effect of change of origin and scale: r is unchanged if each variable is shifted by a constant or multiplied by a positive constant; If the multipliers have opposite signs for the two variables, the sign of r flips but its size stays the same.
Unit-free property: r has no unit; It is a pure number, so it does not depend on the units of measurement.
Definition of r: r = Cov(X, Y) ÷ (σx × σy); σx and σy are population standard deviations (divide by n). Use the same divisor in all three terms.
Covariance: Cov(X, Y) = Σ(X − X̄)(Y − Ȳ) ÷ n = ΣXY ÷ n − X̄ × Ȳ; The second form is faster when ΣXY, X̄ and Ȳ are known.
Direct method (sums): r = [nΣXY − ΣX ΣY] ÷ [√(nΣX² − (ΣX)²) × √(nΣY² − (ΣY)²)]; Needs only five sums and n. No need to find the means.
Deviations from actual means: r = Σxy ÷ √(Σx² × Σy²), where x = X − X̄ and y = Y − Ȳ; Best when the means are whole numbers.
Assumed mean method: r = [nΣdxdy − ΣdxΣdy] ÷ [√(nΣdx² − (Σdx)²) × √(nΣdy² − (Σdy)²)], where dx = X − A and dy = Y − B; A and B are assumed means. Do not confuse the Σdx² terms with (Σdx)².
Step deviation method: u = (X − A) ÷ h, v = (Y − B) ÷ k, then r = r(u, v) using the same formula on u and v; Valid when h and k are both positive. If h and k have opposite signs, r changes sign.
Range and key properties: −1 ≤ r ≤ +1; r(X, Y) = r(Y, X); r is unchanged by change of origin and positive change of scale; If X and Y are independent, r = 0. The converse is not always true.
Coefficient of determination: r² = proportion of variation in one variable explained by a linear relation with the other; For example, r = 0.8 gives r² = 0.64.
Probable error: PE = 0.6745 × (1 − r²) ÷ √n; Commonly used rule: if r < PE, r is not significant. If r > 6 × PE, r is significant.
Spearman's rank correlation (no ties): r = 1 − 6ΣD² ÷ [n(n² − 1)]; D = difference between the two ranks of the same item. n = number of items (pairs).
With tied ranks: r = 1 − 6[ΣD² + Σ m(m² − 1) ÷ 12] ÷ [n(n² − 1)]; m = number of items in each tied group. Add one term for every tied group in either series.
Correction for one tie of 2: m(m² − 1) ÷ 12 = 2 × 3 ÷ 12 = 0.5; A tie of 3 gives 3 × 8 ÷ 12 = 2.
Average rank for a tie: Average rank = (sum of the ranks the tied items would take) ÷ m; The next untied item continues after the full group, so ranks 3 and 4 tied means the next item is rank 5.
Check on rank differences: ΣD = 0; The differences must add up to zero. If not, you made an error in ranking or subtracting.
Range of the value: −1 ≤ r ≤ +1; +1 means identical ranking and −1 means exactly reversed ranking.
Coefficient of concurrent deviation: r_c = ± √( ± (2c − m) ÷ m ); c = number of concurrent deviations (pairs with the same sign). m = number of pairs of deviations = n − 1, where n is the number of pairs of observations.
Number of pairs of deviations: m = n − 1; The first value has no previous value, so it gives no deviation. Never use m = n.
Sign rule: If (2c − m) > 0: r_c = + √((2c − m) ÷ m). If (2c − m) < 0: r_c = − √((m − 2c) ÷ m).; The sign outside the root and inside it is the same as the sign of (2c − m). If 2c = m, r_c = 0.
Direction of each change: + if value rises, − if value falls, 0 if no change; A pair counts in c only when both signs are + or both are −. A pair with a 0 is not counted in c.
Regression line of Y on X: Y − Ȳ = b_yx (X − X̄); Use this to estimate Y when X is given.
Regression line of X on Y: X − X̄ = b_xy (Y − Ȳ); Use this to estimate X when Y is given.
Regression coefficient of Y on X: b_yx = r × σy ÷ σx = (nΣXY − ΣX ΣY) ÷ (nΣX² − (ΣX)²); The denominator uses the X values only.
Regression coefficient of X on Y: b_xy = r × σx ÷ σy = (nΣXY − ΣX ΣY) ÷ (nΣY² − (ΣY)²); The denominator uses the Y values only.
Link with correlation: r² = b_xy × b_yx, so r = ±√(b_xy × b_yx); r has the same sign as both regression coefficients. Both b values always have the same sign.
Point of intersection: Both lines pass through (X̄, Ȳ); Solve the two line equations together to get the means.
Check on the coefficients: b_xy × b_yx ≤ 1; If the product is more than 1, you have assigned the lines the wrong way round.
Regression coefficient of y on x: byx = r × σy ÷ σx = Cov(x, y) ÷ σx²; Slope of the line y − ȳ = byx (x − x̄). Divide by the variance of x.
Regression coefficient of x on y: bxy = r × σx ÷ σy = Cov(x, y) ÷ σy²; Slope of the line x − x̄ = bxy (y − ȳ). Divide by the variance of y.
Link with correlation: r² = byx × bxy, so r = ± √(byx × bxy); Take the sign of the regression coefficients. Both must have the same sign.
Sign property: byx, bxy and r have the same sign; If one coefficient is positive and the other negative, the data is wrong.
Size property: |byx × bxy| ≤ 1; Since |r| ≤ 1. If one coefficient is numerically more than 1, the other must be less than 1.
AM and r: (byx + bxy) ÷ 2 ≥ r; For positive coefficients, the arithmetic mean of the two is at least r (AM ≥ GM). For negative ones, compare absolute values.
Lines meet at the means: Both lines pass through (x̄, ȳ); Solve the two equations together to get x̄ and ȳ.
Change of origin and scale: If u = (x − a) ÷ c and v = (y − b) ÷ d, then byx = (d ÷ c) × bvu and bxy = (c ÷ d) × buv; Regression coefficients do not change with change of origin. They do change with change of scale.
Angle between the regression lines: tan θ = |(1 − r²) ÷ r| × σxσy ÷ (σx² + σy²); If r = 0 the lines are perpendicular. If r = ±1 the lines coincide (θ = 0).

Quick revision

Pearson's r always lies between -1 and +1; r has no unit.
r = Cov(x, y) ÷ (σx × σy), where Cov(x, y) = Σ(x - x̄)(y - ȳ) ÷ n.
r is independent of change of origin and scale in magnitude. If x and y are scaled by factors of opposite sign, r changes sign; if both factors have the same sign, r is unchanged.
Spearman's R = 1 - 6Σd² ÷ [n(n² - 1)], where d is the difference in ranks.
For repeated ranks, add a correction of (m³ - m) ÷ 12 to Σd² for each group of m tied items.
Concurrent deviation: r = ±√(±(2c - n) ÷ n), where c is the number of concurrent deviations and n is the number of pairs of deviations. The sign outside the root is the sign of (2c - n), and (2c - n) ÷ n must be positive for the root to be taken.
Regression of y on x: y - ȳ = byx (x - x̄), with byx = r × σy ÷ σx.
Regression of x on y: x - x̄ = bxy (y - ȳ), with bxy = r × σx ÷ σy.
r² = bxy × byx, so r = ±√(bxy × byx), with the sign of the coefficients.
Both regression lines pass through the point (x̄, ȳ); the means can be found by solving the two line equations together.
Regression coefficients are unaffected by change of origin but are affected by change of scale.
The two regression coefficients have the same sign. Both cannot be greater than 1; if one is greater than 1, the other must be less than 1 (since bxy × byx = r² ≤ 1).

Common mistakes

Thinking r = -0.9 is weaker than r = +0.3. Fix: Compare strength using |r|. The sign only gives direction. So -0.9 is stronger than +0.3.
Saying r = 0 means the variables are unrelated in every way. Fix: Say r = 0 means no linear relationship. A curved relationship may still exist.
Writing Σdx² when the formula needs (Σdx)², or mixing the two up. Fix: Compute them separately and label them: 'sum of squares' and 'square of sum'. In the denominator, use n × (sum of squares) − (square of sum).
Using the wrong n, such as the number of columns or n − 1, or forgetting n in the formula. Fix: Count the pairs of observations. In the sums formula, n multiplies ΣXY, ΣX² and ΣY² only, not the other term in each pair.
Ranking one series highest = 1 and the other lowest = 1. Fix: Pick one direction before you start and apply it to both series. Check ΣD = 0 afterwards.
Giving tied items the same lower rank, such as both 3, instead of 3.5. Fix: Add up the ranks the tied items would take and divide by m. Two items at 3rd and 4th get 3.5 each, and the next item gets 5.
Using m = n instead of m = n − 1. Fix: Always count the signs you actually wrote. The number of signs in one series is m.
Taking the sign of the root wrongly when 2c − m is negative. Fix: Use −√((m − 2c) ÷ m). The number inside the root must be positive, and the sign outside follows 2c − m.
Using the line of Y on X to estimate X. Fix: Ask which variable you are estimating. That is the dependent variable, and its line goes on the left side: Y on X to find Y, X on Y to find X.
Using the wrong denominator for b_xy. Fix: For b_xy both the denominator terms use Y: nΣY² − (ΣY)². The numerator stays the same.

Exam tips

Questions on this topic are mostly conceptual. Read each word of the option, especially 'always', 'only' and 'proves'.
Memorise one real-life example for each type: positive, negative, zero, so you can match options fast.
Check the range -1 to +1 first in every numerical-value option.
Remember r is unaffected by change of origin and by positive scale change. This is a favourite statement-type MCQ.
Skip the question if two options look equally right after elimination, since wrong answers carry 0.25 negative marking.
Questions often give summary sums, so practise the sums formula until the numerator and denominator are quick.
Property questions ask what happens to r after a change like U = aX + b. Positive a: r unchanged. Negative a: sign flips. Practise this logic, because it takes seconds.
Use the sign of the numerator and the range −1 to +1 to remove options before doing long arithmetic.