Quartiles, Deciles & Percentiles – Concepts & Applications 📘📊

Partition values divide the data into equal parts, helping us understand how data is spread and where a particular value stands in comparison to the rest.

They are widely used in:

• Income and salary analysis
• Entrance exams (percentile ranking)
• Market segmentation
• Quality control

The three major types of partition values are:

• Quartiles (Q1, Q2, Q3) → divide data into 4 equal parts
• Deciles (D1–D9) → divide data into 10 parts
• Percentiles (P1–P99) → divide data into 100 parts

1. Quartiles (Q1, Q2, Q3)

Quartiles divide the distribution into four equal parts.

• Q1 (Lower Quartile) = 25% of observations lie below it
• Q2 (Median) = 50% of observations lie below it
• Q3 (Upper Quartile) = 75% of observations lie below it

Formula (Grouped Data)

Qk = L + [( (kN/4) – C.F_prev ) / f ] × h

Where:

• k = 1, 2, or 3
• L = lower limit of the quartile class
• N = total frequency
• C.F_prev = cumulative frequency before quartile class
• f = frequency of quartile class
• h = class width

2. Deciles (D1–D9)

Deciles divide data into ten equal parts.

Formula (Grouped Data)

Dk = L + [( (kN/10) – C.F_prev ) / f ] × h

Where k = 1 to 9.

Example: D5 = Median

3. Percentiles (P1–P99)

Percentiles divide data into 100 equal parts.

Formula (Grouped Data)

Pk = L + [( (kN/100) – C.F_prev ) / f ] × h

Where k = 1 to 99.

Example: P90 indicates the value below which 90% of observations lie.

Visual Understanding

When data is arranged from smallest to largest:

0% —— Q1 —— Q2 —— Q3 —— 100%

0% — D1 — D2 — ... — D9 — 100%

0% — P1 — P2 — ... — P99 — 100%

Quartiles → broader split Deciles → finer split Percentiles → very fine split

Solved Example – Quartiles, Deciles & Percentiles

Given Grouped Data

N = 40

Find Q1

k = 1
kN/4 = 1×40/4 = 10
Q1 lies in C.F. ≥ 10 → class 10–20

Values:

L = 10
C.F_prev = 5
f = 9
h = 10

Q1 = 10 + [(10 – 5) / 9] × 10
   = 10 + (5/9) × 10
   ≈ 10 + 5.56
Q1 ≈ 15.56

Find D3

k = 3
kN/10 = 3×40/10 = 12
D3 lies in C.F. ≥ 12 → class 10–20

Values:

L = 10
C.F_prev = 5
f = 9
h = 10

D3 = 10 + [(12 – 5) / 9] × 10
   = 10 + (7/9) × 10
   ≈ 10 + 7.78
D3 ≈ 17.78

Find P75

k = 75
kN/100 = 75×40/100 = 30
P75 lies in C.F. ≥ 30 → class 30–40

Values:

L = 30
C.F_prev = 26
f = 8
h = 10

P75 = 30 + [(30 – 26) / 8] × 10
    = 30 + (4/8) × 10
    = 30 + 5
P75 = 35

Properties of Partition Values ⭐

1. Positional averages

Do not depend on every value in the dataset.

2. Not affected by extreme values

Useful for skewed distributions.

3. Can be used for open-ended intervals

Especially quartiles and percentiles.

4. Helpful in interpreting distribution spread

Percentiles show relative standing.

5. Quartiles used in box plots (Q1, Median, Q3)

Applications in Business & Economics 📌

• Salary comparisons (median, quartiles)
• Income inequality studies
• Student performance analysis (percentile ranking)
• Identifying market segments
• Understanding dispersion

Summary ✨

• Quartiles → divide data into 4 parts
• Deciles → divide data into 10 parts
• Percentiles → divide data into 100 parts
• Use corresponding formulas for grouped data
• Useful for skewed distributions and rank analysis

Quiz Time 🎯