Theory of Attributes

Unit V · Lecture 38

BDS113 — Introduction to Statistics

Yule's Coefficient of Association Q

Formula · 2×2 Table Notation · Properties · Interpretation · Worked Examples · Comparison with Other Measures

Unit V · Lecture 38 of 40 · B.Sc. Data Science, 1st Year, Semester II

Theory of Attributes

Unit V · Lecture 38

The Journey — Lecture 38

📐

Yule's Q — The Formula
2×2 table notation · Derivation · Intuition behind the formula

🔑

Properties & Interpretation
Range · Special cases · What Q = 0, ±1 mean · Interpretation guide

✏️

Worked Examples & Interactive Calculator
Step-by-step problems · Calculator tool · Quiz

Theory of Attributes

Unit V · Lecture 38

📐 Yule's Coefficient of Association — Q

📖 2×2 Table Setup

The standard 2×2 table for two attributes A and B:

	(B)	(β)
(A)	a	b
(α)	c	d

Where: a = (AB), b = (Aβ), c = (αB), d = (αβ), N = a+b+c+d

📖 Yule's Q Formula

Q = (ad − bc) / (ad + bc)

Named after: George Udny Yule (1871–1951), British statistician.

Intuition: Q compares the product of concordant pairs (ad = both present × both absent) with the product of discordant pairs (bc = A present/B absent × A absent/B present). If concordant pairs dominate (ad > bc), the association is positive.

Theory of Attributes

Unit V · Lecture 38

🔑 Properties of Yule's Q

Range: −1 ≤ Q ≤ +1. Full range from perfect negative to perfect positive association.
Q = +1: Perfect positive association. This happens when b = 0 or c = 0 (one of the off-diagonal cells is zero). Everyone with A also has B, or everyone without A also lacks B.
Q = −1: Perfect negative association. This happens when a = 0 or d = 0. A and B are mutually exclusive or always complementary.
Q = 0: No association (independence). This happens when ad = bc, i.e., (AB)(αβ) = (Aβ)(αB).
Symmetric: Q(A,B) = Q(B,A). The order of attributes doesn't matter.
Unit-free: Q is a pure number with no units.
Only for 2×2 tables: Q is specifically designed for two dichotomous attributes.
Relationship with χ²: Q² ≥ χ²/N (Q is always at least as large as φ in absolute value).

💬 Interpretation Guide:
|Q| > 0.75 → Very strong association
0.50 < |Q| < 0.75 → Strong association
0.25 < |Q| < 0.50 → Moderate association
|Q| < 0.25 → Weak association
Q = 0 → No association (independence)

Theory of Attributes

Unit V · Lecture 38

✏️ Example 1: Computing Yule's Q

📝 Problem

In a study of 500 people, the relationship between smoking (A) and lung disease (B) was recorded:

	Disease (B)	No Disease (β)
Smoker (A)	120	80
Non-smoker (α)	60	240

📖 Solution

a = 120, b = 80, c = 60, d = 240

Q = (ad − bc) / (ad + bc)
   = (120×240 − 80×60) / (120×240 + 80×60)
   = (28800 − 4800) / (28800 + 4800)
   = 24000 / 33600
   = 0.7143

Yule's Coefficient of Association

Q = 0.714

Strong positive association between smoking and lung disease

Smokers are much more likely to have lung disease than non-smokers. Q = 0.714 indicates a strong relationship.

Theory of Attributes

Unit V · Lecture 38

✏️ Example 2: Special Cases of Q

Q = +1 (Perfect Positive)

	B	β
A	100	0
α	50	150

Q = (100×150 − 0×50)/(100×150 + 0×50) = 15000/15000 = +1
Everyone with A also has B (b=0).

Q = −1 (Perfect Negative)

	B	β
A	0	100
α	80	120

Q = (0×120 − 100×80)/(0×120 + 100×80) = −8000/8000 = −1
No one with A has B (a=0). A and B are mutually exclusive.

Q = 0 (Independence)

	B	β
A	50	50
α	50	50

Q = (50×50 − 50×50)/(50×50 + 50×50) = 0/5000 = 0
ad = bc → No association. The proportion of B is the same in both A and α groups.

Theory of Attributes

Unit V · Lecture 38

🎬 Interactive: Yule's Q Calculator

Enter the four cell frequencies:

a = (AB):

b = (Aβ):

c = (αB):

d = (αβ):

Theory of Attributes

Unit V · Lecture 38

🔑 Summary & Quiz — Lecture 38

Yule's Q = (ad − bc) / (ad + bc) where a,b,c,d are the four cells of a 2×2 table.
Range: −1 ≤ Q ≤ +1. Q = +1 (perfect positive), Q = −1 (perfect negative), Q = 0 (independence).
Q = 0 iff ad = bc — this is the condition for independence.
Q is symmetric, unit-free, and designed specifically for 2×2 tables.
Q compares concordant pairs (ad) with discordant pairs (bc).

Q1: Yule's Q formula is:

A) (a+d)/(b+c)

B) (ad−bc)/(ad+bc)

C) (ab−cd)/(ab+cd)

D) (a−d)/(a+d)

Q2: Q = +1 means:

A) No association

B) Perfect positive association

C) Perfect negative association

D) The data is inconsistent

Q3: Q = 0 when:

A) a = d = 0

B) ad = bc

C) a = b = c = d

D) N = 0

Q4: For a=50, b=30, c=20, d=100, Q = ?

A) 0.50

B) 0.724

C) 0.80

D) 1.00

Q5: Yule's Q is applicable to:

A) Any size contingency table

B) Only 2×2 tables

C) Only 3×3 tables

D) Only quantitative data

Theory of Attributes

Unit V · Lecture 38

📝 Homework & Next Lecture

📝 Practice Tasks

1. Compute Q for: a=80, b=40, c=30, d=150. Interpret.

2. Compute Q for: a=25, b=75, c=75, d=25. Interpret.

3. If Q = 0.8, a = 100, b = 20, c = 30, find d.

4. Show that Q = +1 if and only if b = 0 or c = 0.

5. Compare Q and φ for the data in Example 1 (a=120, b=80, c=60, d=240). Which is larger in absolute value?

6. BONUS: Show that Q can be written as Q = (OR − 1)/(OR + 1), where OR = ad/bc is the odds ratio.

Next: Lecture 39

Yule's Coefficient of Colligation Y — An alternative measure with different properties