Introduction to Statistics

Unit III · Lecture 16

BDS113 — Introduction to Statistics

Measures of Dispersion

Introduction to Dispersion · Range · Quartile Deviation

Unit III · Lecture 16 of 40 · B.Sc. Data Science, 1st Year, Semester II

Introduction to Statistics

Unit III · Lecture 16

🤔 Why Do We Need Measures of Dispersion?

💡 The Story of Two Classes

Two classes both have a mean score of 60. Are they equally performing?

	Class A	Class B
Scores	58, 59, 60, 61, 62	20, 40, 60, 80, 100
Mean	60	60
Spread	Very close together	Very far apart

Same mean, completely different behaviour! Class A is consistent. Class B is scattered. The mean alone cannot capture this. We need dispersion.

💬 Key Insight: Central tendency tells us where the centre is. Dispersion tells us how far data spreads around it. Both are needed for a complete picture.

Introduction to Statistics

Unit III · Lecture 16

🎬 Visual: Same Mean, Different Spread

Click each dataset to see how dispersion changes while the mean stays at 60:

Introduction to Statistics

Unit III · Lecture 16

📖 What is Dispersion?

📖 Definition

Dispersion (scatter, spread, variability) refers to the extent to which individual data values differ from each other and from their central value. It quantifies how stretched or squeezed the data is.

💡 Analogy: Archery Targets

Two archers both hit the bullseye on average. Archer A's shots cluster tightly. Archer B's scatter everywhere. Same centre, different precision. Dispersion measures that precision.

📖 Properties of a Good Measure of Dispersion

1. Rigidly defined (not subjective). 2. Easy to calculate. 3. Based on all observations (ideally). 4. Not too sensitive to outliers. 5. Suitable for further algebraic treatment.

Introduction to Statistics

Unit III · Lecture 16

📋 Types of Measures of Dispersion

📏

Absolute Measures — Same units as data. Cannot compare across different units.
Range · Quartile Deviation · Mean Deviation · Standard Deviation

📐

Relative (Coefficient) Measures — Unitless. Can compare datasets with different units.
Coeff. of Range · Coeff. of Q.D. · Coeff. of Variation (C.V.)

Measure	Type	Based On	Lecture
Range	Absolute	Extreme values only	16 (Today)
Quartile Deviation	Absolute	Q₁ and Q₃	16 (Today)
Mean Deviation	Absolute	All observations	17
Std Dev / Variance	Absolute	All observations	18–19
Coeff. of Variation	Relative	Mean & Std Dev	19

Introduction to Statistics

Unit III · Lecture 16

📏 Range

📖 Definition

The range is the simplest measure of dispersion — the difference between the largest (L) and smallest (S) values. It tells us the total end-to-end spread.

Absolute Range

Range = L − S

Coefficient of Range (Relative)

C.R. = (L − S) / (L + S)

Dimensionless. 0 to 1. For comparing datasets.

💡 Analogy: Temperature Range

City A: min 15°C, max 42°C → range = 27°C (wide variation). City B: min 22°C, max 27°C → range = 5°C (stable). Range captures the total swing.

Introduction to Statistics

Unit III · Lecture 16

📝 Range — Worked Examples

Example 1: Ungrouped

Data: 12, 28, 35, 42, 18, 55, 63, 71

📝 Solution

L = 71, S = 12
Range = 71 − 12 = 59
C.R. = 59/83 = 0.711 (high relative dispersion)

Example 2: Grouped

Class	10–20	20–30	30–40	40–50	50–60
f	5	8	12	10	5

📝 Solution

L = 60 (upper boundary), S = 10 (lower boundary)
Range = 60 − 10 = 50
C.R. = 50/70 = 0.714
Note: Approximate — exact min/max unknown within classes.

Introduction to Statistics

Unit III · Lecture 16

✅ Merits & ⚠️ Limitations of Range

✅ Merits

1. Simplest to calculate.
2. Easy to understand.
3. Quick snapshot of total spread.
4. Useful for quality control.
5. C.R. allows comparison.

⚠️ Limitations

1. Based on only 2 values (L, S).
2. Highly affected by outliers.
3. Not algebraically tractable.
4. Cannot handle open-end classes.
5. Unstable across samples.

Introduction to Statistics

Unit III · Lecture 16

🎬 Visual: Range — Seeing the Spread

Introduction to Statistics

Unit III · Lecture 16

📐 Quartile Deviation (Semi-Interquartile Range)

📖 Definition

The Quartile Deviation (Q.D.) is half the IQR. It measures the average spread of the middle 50% of data around the median. Also called Semi-Interquartile Range (SIR).

Quartile Deviation

Q.D. = (Q₃ − Q₁) / 2

Absolute measure. Same units as data.

Coefficient of Q.D.

C.Q.D. = (Q₃ − Q₁) / (Q₃ + Q₁)

Relative. Dimensionless. 0 to 1.

💡 Analogy: The Middle Lane

Q₁ is the left edge, Q₃ is the right edge of the middle lane. IQR = lane width. Q.D. = half the lane width — how far the typical middle observation deviates from the centre line.

Introduction to Statistics

Unit III · Lecture 16

💡 Why Q.D. is Better Than Range

💡 The Outlier Problem

	Dataset X	Dataset Y
Values	10, 20, 30, 40, 50	10, 20, 30, 40, 500
Range	40	490
Q₁, Q₃	15, 45	15, 45
Q.D.	15	15

Range explodes (40→490) from one outlier. Q.D. stays at 15 — robust!

💬 Key Distinction:
Range = spread of entire data (outlier-sensitive).
Q.D. = spread of middle 50% (outlier-robust).
Q.D. ≤ Range always.

Introduction to Statistics

Unit III · Lecture 16

📝 Quartile Deviation — Worked Examples

Example 1: Ungrouped

Data: 12, 18, 22, 25, 28, 31, 35, 38, 42, 48, 55

📝 Solution

n=11. Q₁ = pos 3 = 22. Q₃ = pos 9 = 42.
Q.D. = (42−22)/2 = 10
C.Q.D. = 20/64 = 0.3125

Example 2: Grouped

Class	0–10	10–20	20–30	30–40	40–50	50–60
f	5	8	12	15	7	3

📝 Solution

N=50. CF: 5,13,25,40,47,50.
Q₁: N/4=12.5 → 10–20 → 10+[(7.5/8)×10] = 19.375
Q₃: 3N/4=37.5 → 30–40 → 30+[(12.5/15)×10] = 38.33
Q.D. = (38.33−19.375)/2 = 9.48
C.Q.D. = 18.955/57.705 = 0.328

Introduction to Statistics

Unit III · Lecture 16

🎬 Visual: Q.D. on a Box Plot

Click "Show Q.D." to see how Quartile Deviation relates to the box plot:

Introduction to Statistics

Unit III · Lecture 16

⚖️ Range vs Quartile Deviation

Feature	Range	Quartile Deviation
Formula	L − S	(Q₃ − Q₁) / 2
Based on	2 extreme values	Q₁ and Q₃
Outlier sensitive?	Yes — highly	No — robust
Measures	Total spread	Middle 50% spread
Relative form	(L−S)/(L+S)	(Q₃−Q₁)/(Q₃+Q₁)
Open-end classes?	Cannot	Can be calculated
Best for	Quick overview, QC	Skewed data, outliers

💬 Both are "non-parametric" — they don't use the mean. For the most powerful measure (Standard Deviation), stay tuned for Lectures 18–19!

Introduction to Statistics

Unit III · Lecture 16

🎬 Visual: Interactive Q.D. Calculator

Enter Q₁ and Q₃ to instantly see IQR, Q.D., and C.Q.D.:

Q₁

Q₃

Introduction to Statistics

Unit III · Lecture 16

🔑 Summary — Lecture 16

Dispersion measures how spread out data is. Central tendency alone is insufficient — two datasets with the same mean can have vastly different spreads.
Range = L − S: Simplest, based on extremes. Highly outlier-sensitive. C.R. = (L−S)/(L+S).
Q.D. = (Q₃ − Q₁)/2: Middle 50% spread. Outlier-robust. C.Q.D. = (Q₃−Q₁)/(Q₃+Q₁).
IQR = Q₃ − Q₁. Q.D. = IQR/2. Q.D. ≤ Range always.
Both Range and Q.D. use only 2 values — they don't use all observations. For that, we need Mean Deviation (Lecture 17) or Standard Deviation (Lectures 18–19).
Absolute measures have units. Relative measures (coefficients) are dimensionless for cross-dataset comparison.

Introduction to Statistics

Unit III · Lecture 16

🧠 Quick Quiz — Lecture 16

Q1: The range of 15, 22, 38, 45, 60, 72 is:

A) 57

B) 72

C) 15

D) 47

Q2: Coefficient of Range for L=80, S=20 is:

A) 60

B) 0.60

C) 0.40

D) 0.75

Q3: Quartile Deviation is also called:

A) Interquartile Range

B) Semi-Interquartile Range

C) Quartile Range

D) Median Deviation

Q4: If Q₁=25, Q₃=45, the Q.D. is:

A) 45

B) 10

C) 20

D) 25

Q5: Which is NOT affected by outliers?

A) Range

B) Quartile Deviation

C) Both affected

D) Neither affected

Q6: Main limitation of range:

A) Hard to calculate

B) Depends on only two extreme values

C) Cannot be calculated for grouped data

D) Always equals zero

Q7: C.Q.D. for Q₁=30, Q₃=50:

A) 0.20

B) 0.25

C) 0.40

D) 0.50

Q8: IQR and Q.D. relationship:

A) IQR = Q.D.

B) IQR = 2 × Q.D.

C) Q.D. = 2 × IQR

D) IQR = Q.D.²

Q9: An absolute measure of dispersion has:

A) No units

B) Same units as data

C) Percentage units

D) Squared units

Q10: For open-end distributions, which can be calculated?

A) Range

B) Quartile Deviation

C) Both

D) Neither

Introduction to Statistics

Unit III · Lecture 16

📝 Homework & Practice

📝 Practice Tasks

1. Find range and C.R. for: (a) 14,28,35,42,56,63,71,88 (b) 5.2,6.8,7.1,8.5,9.3,10.1

2. Find range for grouped data:

Class	20–30	30–40	40–50	50–60	60–70
f	6	10	14	12	8

3. Find Q.D. and C.Q.D. for: 8,12,18,25,30,36,42,48,55,60,68

4. Find Q.D. and C.Q.D. from grouped data in Q2.

5. Two datasets have mean=50. A has range 10, B has range 80. What can you conclude?

6. Range=48, C.R.=0.6. Find L and S.

7. Why is Q.D. preferred over range with outliers? Give a numerical example.

8. BONUS: For symmetric data, Q₁=30, Q₃=50. Verify median=40. Find Q.D. and C.Q.D.

Introduction to Statistics

Unit III · Lecture 16

Thank You!

Coming Up in Unit III

Lecture 17: Mean Deviation
Lectures 18–19: Standard Deviation & Variance
Lecture 20: Combined S.D. & Coefficient of Variation
Lectures 21–22: Skewness & Kurtosis
Lecture 23: Unit III Review

Unit III · Lecture 16 · BDS113 · B.Sc. Data Science, Semester II