Regression and estimation

Business Statistics · Unit IV · Topic 12 of 15 · 5 min read

Meaning of regression, the two lines and regression coefficients, their properties, correlation versus regression, and regression equations from raw data and from means, SDs and r, with estimation, worked on the 2024–2026 papers.

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On this page

1. Meaning
2. The two regression lines
3. Correlation versus regression (January 2025 short note)
4. Worked: raw data (January 2026)
5. Worked: from means, SDs and r
How this is examined

Cases checked to exist. Read the judgment before quoting a holding.

Every case cited here has been checked to exist — the citation resolves and names the right parties. The propositions attributed to each case have not been verified line by line against the judgment. Case names link through to the full judgment: read it before quoting a holding in an examination answer, and never carry a proposition from notes into court on trust.

The examinable core. Two lines, always: Y on X (to estimate Y for a given X) and X on Y (to estimate X for a given Y). From summary data, b_yx = r σ_y/σ_x and b_xy = r σ_x/σ_y. To estimate X when Y is given, use the X on Y line, never the other one rearranged.

1. Meaning

Regression is the estimation of the unknown value of one variable from the known value of another, on the basis of the average relationship between them. The term was used by Sir Francis Galton, who found that the heights of sons of unusually tall or short fathers tend to “regress” towards the average height. Correlation tells us whether and how closely two variables are related; regression tells us the form of the relationship and lets us predict.

The variable estimated is the dependent (explained) variable; the one used to estimate it is the independent (explanatory) variable.

2. The two regression lines

A regression line is the line of best fit by the method of least squares: it makes the sum of squared vertical (or horizontal) deviations of the points from the line a minimum.

Y on X: Y − ȳ = b_yx(X − x̄), with b_yx = r σ_y/σ_x = Σxy / Σx² (x, y deviations from actual means).
X on Y: X − x̄ = b_xy(Y − ȳ), with b_xy = r σ_x/σ_y = Σxy / Σy².

Why two lines? Minimising errors in Y gives one line; minimising errors in X gives another. They coincide only when r = ±1; when r = 0 they are at right angles.

Properties of regression coefficients: (1) r = ±√(b_yx × b_xy), the sign being that of the coefficients; (2) both coefficients have the same sign, and r takes that sign; (3) if one coefficient exceeds 1, the other must be less than 1, since their product r² ≤ 1; (4) the two lines intersect at (x̄, ȳ); (5) the coefficients are independent of change of origin but not of scale.

3. Correlation versus regression (January 2025 short note)

Point	Correlation	Regression
Purpose	Degree and direction of relationship	Nature of relationship; prediction
Variables	Both treated alike	One dependent, one independent
Coefficient	r is symmetric (r_xy = r_yx)	b_yx ≠ b_xy in general
Range	−1 to +1	Any value; one may exceed 1
Units	Pure number	In units of the dependent variable per unit of the other
Origin and scale	Independent of both	Independent of origin, not of scale
Nonsense results	Spurious correlation possible	No such concept; it assumes a cause-and-effect direction

4. Worked: raw data (January 2026)

X: 1, 2, 3, 4, 5. Y: 2, 3, 5, 4, 6. x̄ = 15/5 = 3; ȳ = 20/5 = 4.

X	Y	x = X − 3	y = Y − 4	x²	y²	xy
1	2	−2	−2	4	4	4
2	3	−1	−1	1	1	1
3	5	0	1	0	1	0
4	4	1	0	1	0	0
5	6	2	2	4	4	4
15	20	0	0	10	10	9

b_yx = 9/10 = 0.9; b_xy = 9/10 = 0.9.
Y on X: Y − 4 = 0.9(X − 3) ⇒ Y = 0.9X + 1.3.
X on Y: X − 3 = 0.9(Y − 4) ⇒ X = 0.9Y − 0.6.
Check: r = √(0.9 × 0.9) = 0.9.

August 2024 (X: 1, 5, 3, 2, 1, 2, 7, 3; Y: 6, 1, 0, 0, 1, 2, 1, 5): x̄ = 3, ȳ = 2, Σx² = 30, Σy² = 36, Σxy = −10. b_yx = −10/30 = −0.333, b_xy = −10/36 = −0.278. Y = 3 − 0.333X; X = 3.556 − 0.278Y; r = −√(0.0926) = −0.30.

5. Worked: from means, SDs and r

June 2025: x̄ = 15.5, ȳ = 22, σ_x = 3, σ_y = 4, r = 0.85.

b_yx = 0.85 × 4/3 = 1.1333; Y − 22 = 1.1333(X − 15.5) ⇒ Y = 1.1333X + 4.433.
b_xy = 0.85 × 3/4 = 0.6375; X − 15.5 = 0.6375(Y − 22) ⇒ X = 0.6375Y + 1.475.

January 2025: x̄ = 25, ȳ = 30, σ_x = 5, σ_y = 4, r = 0.8. Estimate X when Y = 20.

b_yx = 0.8 × 4/5 = 0.64; Y − 30 = 0.64(X − 25) ⇒ Y = 0.64X + 14.
b_xy = 0.8 × 5/4 = 1.0; X − 25 = 1.0(Y − 30) ⇒ X = Y − 5.
When Y = 20: X = 20 − 5 = 15.

August 2024: x̄ = 20, ȳ = 25, variance of X = 4, variance of Y = 9, r = 0.75. Here σ_x = √4 = 2 and σ_y = √9 = 3.

b_yx = 0.75 × 3/2 = 1.125; Y = 1.125X + 2.5.
b_xy = 0.75 × 2/3 = 0.5; X = 0.5Y + 7.5.

Trap. When the paper gives variances (August 2024), take square roots before using r σ_y/σ_x. Using 9/4 instead of 3/2 gives b_yx = 1.69 and a wrong line.

How this is examined

Regression equations from raw data: January 2026 and August 2024 (ten marks). From means, SDs and r: June 2025, January 2025 (with an estimate of X when Y = 20) and August 2024 (variances given), all six marks. Theory: “What is regression? Explain the lines of regression” (January 2026), “Write a short note on regression analysis” (June 2025) and “Difference between correlation and regression” (January 2025).

Now try the real questions: Business Statistics past question papers.