Introduction to Statistics

Unit II · Lectures 13–14

BDS113 — Introduction to Statistics

Partition Values

Quartiles · Deciles · Percentiles — Dividing Data into Equal Parts

Unit II · Lectures 13–14 of 40 · B.Sc. Data Science, 1st Year, Semester II

Introduction to Statistics

Unit II · Lectures 13–14

The Journey — Lectures 13 & 14

📐

Lecture 13 — Quartiles
Partition values concept · Quartiles (Q₁, Q₂, Q₃) ungrouped & grouped · IQR · Quartile Deviation · Coefficient of Q.D.

📊

Lecture 14 — Deciles & Percentiles
Deciles (D₁–D₉) ungrouped & grouped · Percentiles (P₁–P₉₉) ungrouped & grouped · Equivalences · Box Plot · Applications

Introduction to Statistics

Unit II · Lecture 13

📏 What Are Partition Values?

📖 Definition

Partition values (also called quantiles or fractiles) are values that divide an ordered dataset into equal parts. They are positional measures. The median is itself a partition value — it divides data into two equal halves.

💡 Analogy: Cutting a Rope

Imagine a rope of length 100 cm. You place marks at specific points to cut it into equal pieces. 1 cut → median. 3 cuts → quartiles. 9 cuts → deciles. 99 cuts → percentiles.

½
Median
2 equal parts · 1 cut

¼
Quartiles
4 equal parts · 3 cuts

⅒
Deciles
10 equal parts · 9 cuts

¹⁄₁₀₀
Percentiles
100 equal parts · 99 cuts

Introduction to Statistics

Unit II · Lecture 13

🎬 Visual: Partition Values on a Number Line

Select which partition values to display:

Introduction to Statistics

Unit II · Lecture 13

📐 Quartiles

📖 Definition

Quartiles are values that divide an ordered dataset into four equal parts. There are three quartiles:

• Q₁ (First / Lower Quartile): 25% below. Position = N/4.
• Q₂ (Second Quartile / Median): 50% below. Position = N/2. Q₂ = Median.
• Q₃ (Third / Upper Quartile): 75% below. Position = 3N/4.

💡 Analogy: Dividing a Cake into 4

Q₁ separates the first quarter. Q₂ (median) splits the halves. Q₃ separates the last quarter. Between Q₁ and Q₃ lies the middle 50% of data.

💬 Key Insight: Q₂ = Median. The IQR = Q₃ − Q₁ captures the spread of the middle 50%, making it robust against outliers.

Introduction to Statistics

Unit II · Lecture 13

🎬 Visual: Quartiles — Dividing Data into Four Parts

Click "Show Quartiles" to see how Q₁, Q₂, Q₃ divide 20 data points into four groups:

Introduction to Statistics

Unit II · Lecture 13

📝 Quartiles — Ungrouped Data

Formulas — Ungrouped Data

Q₁ = Value at position (n+1)/4

Q₂ = Median = Value at position (n+1)/2

Q₃ = Value at position 3(n+1)/4

Arrange data ascending. If position is fractional, interpolate.

📝 Example

Find Q₁, Q₂, Q₃ for: 12, 18, 22, 25, 28, 31, 35, 38, 42, 48, 55

📝 Solution

n = 11. Data is ordered.

Q₁ = position (11+1)/4 = 3rd value = 22
Q₂ = position (11+1)/2 = 6th value = 31 (= Median)
Q₃ = position 3(11+1)/4 = 9th value = 42

IQR = Q₃ − Q₁ = 42 − 22 = 20

📊 Interpretation

• 25% below 22 (Q₁) · 50% below 31 (Q₂) · 75% below 42 (Q₃)
• The middle 50% lies between 22 and 42 (IQR = 20)

Introduction to Statistics

Unit II · Lecture 13

🎬 Visual: Quartile Calculator — Ungrouped Data

Click "Calculate" to watch Q₁, Q₂, Q₃ being identified position by position:

Introduction to Statistics

Unit II · Lecture 13

📊 Quartiles — Grouped (Frequency) Data

📖 General Formula for Partition Values in Grouped Data

Same pattern as the median formula — only the position changes.

Quartile Formula — Grouped Data

Qₖ = L + [(kN/4 − CF) / f] × h

k = 1, 2, or 3 · L = lower limit of quartile class · N = total frequency
CF = CF of class before quartile class · f = frequency of quartile class · h = class width

💡 Q₂ Check

Q₂ = L + [(2N/4 − CF)/f] × h = L + [(N/2 − CF)/f] × h — identical to the median formula. Q₂ = Median always.

Introduction to Statistics

Unit II · Lecture 13

📝 Quartiles — Grouped Data: Worked Example

📝 Problem

Find Q₁, Q₂, Q₃:

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

📝 Step-by-Step Solution

N = 50. CF: 5, 13, 25, 40, 47, 50.

Q₁ (k=1): N/4 = 12.5. CF first ≥ 12.5 at 10–20 (CF=13).
L=10, CF(before)=5, f=8, h=10
Q₁ = 10 + [(12.5−5)/8] × 10 = 10 + 9.375 = 19.375

Q₂ (k=2): N/2 = 25. CF first ≥ 25 at 20–30 (CF=25).
L=20, CF(before)=13, f=12, h=10
Q₂ = 20 + [(25−13)/12] × 10 = 20 + 10 = 30

Q₃ (k=3): 3N/4 = 37.5. CF first ≥ 37.5 at 30–40 (CF=40).
L=30, CF(before)=25, f=15, h=10
Q₃ = 30 + [(37.5−25)/15] × 10 = 30 + 8.33 = 38.33

IQR = 38.33 − 19.375 = 18.96

Introduction to Statistics

Unit II · Lecture 13

📐 Quartile Deviation (Semi-Interquartile Range)

Quartile Deviation (Q.D.)

Q.D. = (Q₃ − Q₁) / 2

Average spread of the middle 50% around the median.

Coefficient of Quartile Deviation

C.Q.D. = (Q₃ − Q₁) / (Q₃ + Q₁)

Dimensionless relative measure. Ranges 0 to 1. Good for comparing datasets.

📝 From Our Example

Q₁ = 19.375, Q₃ = 38.33

Q.D. = (38.33 − 19.375) / 2 = 9.48

C.Q.D. = 18.955 / 57.705 = 0.328

💡 Interpretation: Q.D. = 9.48 means the middle 50% deviates ±9.48 from the median on average. C.Q.D. = 0.328 indicates moderate dispersion.

Introduction to Statistics

Unit II · Lecture 14

— Lecture 14 —

Deciles & Percentiles

D₁–D₉ · P₁–P₉₉ · Relationships · Applications

Introduction to Statistics

Unit II · Lecture 14

📏 Deciles

📖 Definition

Deciles divide data into 10 equal parts. Nine deciles: D₁–D₉. Dₖ has 10k% below it.

Ungrouped Data

Dₖ = Value at position k(n+1)/10

Grouped Data

Dₖ = L + [(kN/10 − CF) / f] × h

💡 Understanding Deciles

• D₁ → 10% below
• D₅ → 50% below → D₅ = Median = Q₂
• D₉ → 90% below

Use: "Top 10% earners" starts at D₉. "Below 3rd decile" means below D₃.

Introduction to Statistics

Unit II · Lecture 14

📝 Deciles — Worked Example

📝 Ungrouped: Find D₃, D₇

Data: 15, 22, 28, 33, 38, 42, 47, 53, 60, 68 (n=10)

📝 Solution

D₃ = position 3(11)/10 = 3.3rd
= 28 + 0.3(33−28) = 28 + 1.5 = 29.5

D₇ = position 7(11)/10 = 7.7th
= 47 + 0.7(53−47) = 47 + 4.2 = 51.2

📝 Grouped: Find D₃

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

📝 Solution

N=50. kN/10 = 15. CF: 6, 16, 30, 42, 50.
CF first ≥ 15 at 20–30 (CF=16).
L=20, CF(before)=6, f=10, h=10
D₃ = 20 + [(15−6)/10]×10 = 20 + 9 = 29

Introduction to Statistics

Unit II · Lecture 14

📏 Percentiles

📖 Definition

Percentiles divide data into 100 equal parts. 99 percentiles: P₁–P₉₉. Pₖ has k% below it. Most fine-grained partition values.

Ungrouped Data

Pₖ = Value at position k(n+1)/100

Grouped Data

Pₖ = L + [(kN/100 − CF) / f] × h

💡 Real-Life Applications

• 99th percentile in exams → scored better than 99%
• Growth charts: child at 75th percentile is taller than 75% of peers
• Outlier detection: values below P₁ or above P₉₉ are unusual
• Poverty lines: below P₂₀ might be "below poverty line"

Introduction to Statistics

Unit II · Lecture 14

📝 Percentiles — Worked Example

📝 Problem

Find P₂₅, P₅₀, P₇₅:

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.

P₂₅: kN/100 = 12.5 → same as Q₁ → 19.375
P₅₀: kN/100 = 25 → same as Q₂ → 30
P₇₅: kN/100 = 37.5 → same as Q₃ → 38.33

💡 Not a coincidence! P₂₅ = Q₁, P₅₀ = Q₂ = Median, P₇₅ = Q₃. This is the fundamental relationship between quartiles and percentiles.

Introduction to Statistics

Unit II · Lecture 14

🔗 The Master Relationship

Fundamental Equivalences

Q₁ = D₂ = P₂₅
Q₂ = D₅ = P₅₀ = Median
Q₃ = D₈ = P₇₅

Quartile Qₖ = Decile D₂ₖ = Percentile P₂₅ₖ
Decile Dₖ = Percentile P₁₀ₖ

% Below	Quartile	Decile	Percentile
10%	—	D₁	P₁₀
25%	Q₁	D₂	P₂₅
50%	Q₂ (Median)	D₅	P₅₀
75%	Q₃	D₈	P₇₅
90%	—	D₉	P₉₀

💬 Exam Tip: If asked "Find P₂₅" and you already have Q₁, just write P₂₅ = Q₁. These equivalences save time and are tested frequently.

Introduction to Statistics

Unit II · Lecture 14

🎬 Visual: All Partition Values — One Scale

Introduction to Statistics

Unit II · Lecture 14

⚖️ Quartiles vs Deciles vs Percentiles

Feature	Quartiles	Deciles	Percentiles
Parts	4 equal	10 equal	100 equal
Count	3 (Q₁,Q₂,Q₃)	9 (D₁–D₉)	99 (P₁–P₉₉)
Ungrouped	k(n+1)/4	k(n+1)/10	k(n+1)/100
Grouped	L+[(kN/4−CF)/f]×h	L+[(kN/10−CF)/f]×h	L+[(kN/100−CF)/f]×h
Median link	Q₂ = Median	D₅ = Median	P₅₀ = Median
Common use	Box plots, IQR	Income analysis	Exam ranks, growth charts

💡 Unifying Formula: All partition values share Value = L + [(Position − CF)/f] × h. Only Position changes: N/4, kN/10, kN/100.

Introduction to Statistics

Unit II · Lecture 14

📦 Application: Box Plot (Box-and-Whisker)

📖 What is a Box Plot?

A box plot visualises data using five key values: Min, Q₁, Median, Q₃, Max. One of the most powerful EDA tools — built entirely on quartiles.

Introduction to Statistics

Unit II · Lecture 14

🎬 Visual: Partition Value Calculator

Select type, enter k, and see the position formula:

n =

Introduction to Statistics

Unit II · Lectures 13–14

🔑 Complete Summary: Lectures 13–14

Partition values divide ordered data into equal parts: Quartiles (4 parts), Deciles (10 parts), Percentiles (100 parts).
Quartiles: Q₁(25%), Q₂(50%=Median), Q₃(75%). Ungrouped: k(n+1)/4. Grouped: Qₖ = L + [(kN/4 − CF)/f] × h.
IQR = Q₃ − Q₁. Q.D. = (Q₃−Q₁)/2. C.Q.D. = (Q₃−Q₁)/(Q₃+Q₁).
Deciles: D₁–D₉. D₅ = Median. Ungrouped: k(n+1)/10. Grouped: Dₖ = L + [(kN/10 − CF)/f] × h.
Percentiles: P₁–P₉₉. P₅₀ = Median. Ungrouped: k(n+1)/100. Grouped: Pₖ = L + [(kN/100 − CF)/f] × h.
Equivalences: Q₁ = D₂ = P₂₅, Q₂ = D₅ = P₅₀ = Median, Q₃ = D₈ = P₇₅.
Box plot uses Min, Q₁, Median, Q₃, Max to visualise spread and symmetry.
Applications: Quartiles → box plots/IQR. Deciles → income analysis. Percentiles → exam ranks/growth charts.

Introduction to Statistics

Unit II · Lectures 13–14

🧠 Quick Quiz — Lectures 13–14

Q1: Q₂ is the same as:

A) Q₁

B) Median

C) Q₃

D) Mode

Q2: 75% of observations lie below:

A) Q₁

B) D₅

C) Q₃

D) P₂₅

Q3: D₅ equals:

A) Q₁

B) Median

C) P₂₅

D) Q₃

Q4: Quartile Deviation =:

A) Q₃ − Q₁

B) (Q₃ − Q₁) / 2

C) (Q₃ + Q₁) / 2

D) Q₃ / Q₁

Q5: P₂₅ is equivalent to:

A) Q₁ and D₂

B) Q₂ and D₅

C) Q₃ and D₈

D) Median only

Q6: For n=20, the position of D₇ is:

A) 14.0

B) 14.7

C) 15.0

D) 7.0

Q7: A box plot uses:

A) Mean, Median, Mode, Min, Max

B) Min, Q₁, Median, Q₃, Max

C) Q₁, Q₂, Q₃, D₁, D₉

D) P₁₀, P₂₅, P₅₀, P₇₅, P₉₀

Q8: IQR represents:

A) Full range of data

B) Spread of middle 50%

C) Distance from mean to median

D) Range of lowest 25%

Q9: "99th percentile" means:

A) Scored 99 marks

B) Scored better than 99% of test-takers

C) 1% scored above you

D) Both B and C

Q10: The general formula for all partition values in grouped data is:

A) L + [(Position − CF) / f] × h

B) (Position × f) / h + L

C) L + f × h / Position

D) CF + (N × h) / f

Introduction to Statistics

Unit II · Lectures 13–14

📝 Homework & Practice

📝 Practice Tasks

1. Find Q₁, Q₂, Q₃, IQR for: 14, 22, 28, 33, 38, 42, 48, 55, 62, 70, 78.

2. Find Q₁ and Q₃ from grouped data and compute Q.D. and C.Q.D.:

Class	10–20	20–30	30–40	40–50	50–60	60–70
f	6	10	14	18	8	4

3. Find D₃, D₅, D₉ for: 8, 12, 18, 25, 30, 36, 42, 48, 55, 60.

4. Find P₁₀, P₂₅, P₅₀, P₇₅, P₉₀ from the grouped data in Q2.

5. Verify: (a) P₂₅ = Q₁ (b) P₅₀ = Median (c) P₇₅ = Q₃ using data in Q2.

6. A student at the 85th percentile among 200 participants. How many scored below?

7. Draw a box plot for Q1 data. Label all five key values.

8. Why is IQR better than range for data with outliers?

9. BONUS: In 500 values, Q₁=25, D₇=65, P₉₅=88. What % lies between 25 and 65? Between 65 and 88?

Introduction to Statistics

Unit II · Lectures 13–14

Thank You!

Questions? Let's discuss!

Upcoming in Unit II

Lecture 15: Unit II Review & Problem-Solving Session

Then: Unit III — Measures of Dispersion, Skewness & Kurtosis

Unit II · Lectures 13–14 · BDS113 Introduction to Statistics · B.Sc. Data Science, Semester II