Unit 3: Correlation & regression analysis
Business Statistics notes · PTU syllabus (BCOMGE 201-18)
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Unit summary
Businesses often ask how two variables move together — advertising and sales, price and demand. Correlation measures the strength of the relationship; regression predicts one variable from another. This unit covers types of correlation, scatter diagrams, Pearson's and rank correlation, and regression using the least squares method.
After this unit you can
- Explain types of correlation and draw scatter diagrams
- Calculate Karl Pearson's coefficient of correlation
- Calculate Spearman's rank correlation
- Fit regression lines by least squares and state the properties of regression coefficients
PTU syllabus topics
- Meaning of correlation — simple
- multiple
- partial
- linear and non-linear
- scatter diagram
- Pearson's correlation coefficient
- Rank correlation
- principle of least squares
- regression coefficient and its properties
- relationship between correlation and regression
Karl Pearson's r
Cov(x, y) / (σx σy)
Ranges from −1 to +1
Spearman's rank
1 − 6Σd² / [n(n² − 1)]
Regression of y on x
y − ȳ = byx (x − x̄)
Link
r = ±√(byx × bxy)
Topic 1
Correlation and its types
Correlation measures the degree of relationship between two variables.
Positive
Both move in the same direction
Advertising and sales
Negative
Move in opposite directions
Price and demand
Simple vs multiple
Two variables vs three or more
Yield on rainfall vs on rainfall and fertiliser
Partial
Two variables, holding others constant
Yield and rainfall with temperature fixed
Linear vs non-linear
Constant vs changing ratio of change
Straight line vs curve
A scatter diagram plots pairs of values; the pattern of dots shows the direction and strength of the relationship.
Topic 2
Karl Pearson's coefficient of correlation
r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² × Σ(y − ȳ)²] r lies between −1 and +1: +1 perfect positive, −1 perfect negative, 0 no linear correlation.
Example
x: 1, 2, 3, 4, 5 and y: 2, 4, 5, 4, 5. With x̄ = 3 and ȳ = 4: Σdxdy = 6, Σdx² = 10, Σdy² = 6, so r = 6/√60 ≈ 0.77 — fairly high positive correlation.
The probable error PE = 0.6745 (1 − r²)/√n; if r > 6 PE, correlation is significant.
Topic 3
Spearman's rank correlation
Used when data are ranks or qualitative (beauty, honesty): R = 1 − 6Σd² / [n(n² − 1)], where d is the difference in ranks.
Example
Ranks of 5 contestants by two judges give Σd² = 8: R = 1 − 48/120 = 0.6.
Topic 4
Regression analysis
Regression estimates the value of one variable from another. Two regression lines exist:
- y on x: y − ȳ = byx (x − x̄), with byx = r × σy/σx
- x on y: x − x̄ = bxy (y − ȳ), with bxy = r × σx/σy
The principle of least squares fits the line that minimises the sum of squared vertical deviations.
- Geometric mean
- r = ±√(byx × bxy)
- Same sign
- byx, bxy and r all have the same sign
- Both cannot exceed 1
- If one is above 1, the other is below 1
- Independent of origin
- But not of scale
- Lines meet
- At (x̄, ȳ)
Purpose
Measures strength of relationship
Predicts one variable from another
Symmetry
rxy = ryx
byx ≠ bxy
Cause and effect
Not implied
Treats one as dependent
Key terms
- Correlation
- Degree of relationship between variables
- Scatter diagram
- A plot of paired values
- Rank correlation
- Correlation between ranks
- Regression
- Estimating one variable from another
- Least squares
- Method minimising the sum of squared errors
Quick revision
- −1 ≤ r ≤ +1.
- R = 1 − 6Σd²/[n(n² − 1)].
- byx = r σy/σx; bxy = r σx/σy; r = ±√(byx bxy).
- Regression lines meet at the means.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Define correlation and give its types.
- Q2.What does r = −1 indicate?
- Q3.When is rank correlation used?
- Q4.Differentiate between correlation and regression.
- Q5.State two properties of regression coefficients.
Long-answer questions
- Q1.Calculate Karl Pearson's coefficient of correlation for given data and interpret it.
- Q2.Calculate Spearman's rank correlation, including tied ranks.
- Q3.Fit both regression lines and estimate y when x = a given value.
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