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Unit 2 of 4 · M.Com Sem 1

Unit 2: Correlation, regression and probability

Quantitative Techniques notes · PTU syllabus (MCOP103-18)

3 min read7 topics10 exam questions
On this page
  1. Unit summary
  2. Correlation and its types
  3. Karl Pearson's coefficient of correlation
  4. Spearman's rank correlation
  5. Regression analysis
  6. Approaches to probability
  7. Laws of probability
  8. Bayes' theorem
  9. Key terms
  10. Quick revision
  11. Important questions

Unit summary

Business variables move together, and decisions under uncertainty require probability. This unit covers simple, multiple and partial correlation, linear and non-linear correlation, scatter diagrams, Pearson's and rank correlation, simple regression by least squares, the relationship between correlation and regression, approaches to probability, the addition and multiplication laws, conditional probability and Bayes' theorem.

After this unit you can

  • Explain types of correlation and compute Pearson's and rank correlation
  • Fit regression lines by least squares and explain their properties
  • Explain approaches to probability and apply the laws of probability
  • Apply conditional probability and Bayes' theorem

PTU syllabus topics

  • Simple/multiple/partial and linear/non-linear correlation
  • scatter diagrams
  • Pearson's correlation coefficient
  • rank correlation
  • simple regression via least squares
  • relationship between correlation and regression
  • approaches to probability
  • addition and multiplication laws
  • conditional probability and Bayes' theorem
Key formulasCorrelation, regression and probability

  • Pearson's r

    Cov(x, y) / (σx σy)

  • Spearman's rank

    1 − 6Σd² / [n(n² − 1)]

  • Regression coefficient

    byx = r × σy / σx

  • Bayes' theorem

    P(Ai given B) = P(Ai) P(B given Ai) / Σ P(Aj) P(B given Aj)

1

Topic 1

Correlation and its types

Correlation measures the degree of relationship between two variables.

ComparisonTypes of correlation
Meaning
Example

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.

  • Simple correlation: two variables. Multiple correlation: one variable with two or more others jointly (R1.23). Partial correlation: two variables with the effect of others held constant (r12.3).
Key formulasPartial and multiple correlation (three variables)

  • Partial correlation

    r12.3 = (r12 − r13 r23) ÷ √[(1 − r13²)(1 − r23²)]

  • Multiple correlation

    R1.23 = √[(r12² + r13² − 2 r12 r13 r23) ÷ (1 − r23²)]

2

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.

3

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.

4

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.

Key termsProperties of regression coefficients
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)
ComparisonCorrelation vs regression
Correlation
Regression

Purpose

Measures strength of relationship

Predicts one variable from another

Symmetry

rxy = ryx

byx ≠ bxy

Cause and effect

Not implied

Treats one as dependent

5

Topic 5

Approaches to probability

Probability is a number between 0 and 1 measuring the chance that an event will occur.

  • Classical: P(A) = favourable outcomes / total equally likely outcomes.
  • Relative frequency (empirical): P(A) = number of times A occurred / number of trials.
  • Subjective: based on personal judgement and experience.
  • Axiomatic: based on axioms (Kolmogorov) — probability is non-negative, P(sample space) = 1, and additive for mutually exclusive events.
6

Topic 6

Laws of probability

Key formulasLaws of probability

  • Addition law

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

  • Mutually exclusive events

    P(A ∪ B) = P(A) + P(B)

  • Multiplication law

    P(A ∩ B) = P(A) × P(B given A)

  • Independent events

    P(A ∩ B) = P(A) × P(B)

  • Conditional probability

    P(A given B) = P(A ∩ B) / P(B)

Example

A card is drawn from a pack of 52. P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

7

Topic 7

Bayes' theorem

Bayes' theorem revises probabilities when new information arrives: P(Aᵢ given B) = P(Aᵢ) P(B given Aᵢ) / Σ P(Aⱼ) P(B given Aⱼ)

Example

Machines A and B make 60% and 40% of output, with defect rates 2% and 5%. A defective item is found. P(from A) = (0.6 × 0.02) / (0.6 × 0.02 + 0.4 × 0.05) = 0.012/0.032 = 0.375.

Key terms

Partial correlation
Correlation between two variables with others held constant
Regression coefficient
Change in the dependent variable per unit change in the independent variable
Least squares
Method minimising the sum of squared deviations
Conditional probability
Probability of an event given that another has occurred
Bayes' theorem
Revising probabilities using new information

Quick revision

  • r between −1 and +1; rank correlation for qualitative data.
  • Partial r12.3 and multiple R1.23 formulas.
  • Regression lines y on x and x on y meet at the means; r = ±√(byx × bxy).
  • Addition law P(A ∪ B) = P(A) + P(B) − P(A ∩ B); multiplication P(A ∩ B) = P(A) P(B|A).
  • Bayes: posterior ∝ prior × likelihood.

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

  1. Q1.Distinguish simple, partial and multiple correlation.
  2. Q2.What is a scatter diagram?
  3. Q3.State two properties of regression coefficients.
  4. Q4.Distinguish correlation and regression.
  5. Q5.State the addition theorem of probability.
  6. Q6.What is Bayes' theorem?

Long-answer questions

  1. Q1.Explain the types of correlation and compute Pearson's coefficient.
  2. Q2.Explain rank correlation with an example.
  3. Q3.Explain regression analysis and the relationship between correlation and regression.
  4. Q4.Explain the laws of probability, conditional probability and Bayes' theorem.

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