Measures of Shape

Unit III · Lecture 22

BDS113 — Introduction to Statistics

Measures of Kurtosis

Peakedness & Tail Behaviour · β₂ and γ₂ · Types of Kurtosis · Interpretation & Applications

Unit III · Lecture 22 of 40 · B.Sc. Data Science, 1st Year, Semester II

Measures of Shape

Unit III · Lecture 22

The Journey — Lecture 22

📐

What is Kurtosis?
Concept · Peakedness vs tail heaviness · Visual intuition

📊

Karl Pearson's β₂ and Excess Kurtosis γ₂
Formulas · Mesokurtic, leptokurtic, platykurtic · Worked examples

🔗

Percentile-Based Kurtosis & Applications
Percentile kurtosis · Real-world applications · Finance, quality control

⚖️

Properties, Comparisons & Quiz
Properties · Kurtosis vs skewness · Interactive tools · Practice quiz

Measures of Shape

Unit III · Lecture 22

🤔 What is Kurtosis?

📖 Formal Definition

Kurtosis is a statistical measure that describes the shape of a distribution's tails relative to its overall shape. Specifically, it measures whether data is concentrated in the tails (heavy-tailed) or concentrated near the centre (light-tailed) compared to a normal distribution.

💡 Analogy: Hats on a Hat Rack

Think of different distributions as different types of hats sitting on a hat rack:

Leptokurtic (high kurtosis) = Top hat — tall, narrow peak with long brims (heavy tails). Most data is either very close to the centre or very far away.

Mesokurtic (normal kurtosis) = Baseball cap — moderate peak, moderate brim. The "standard" shape.

Platykurtic (low kurtosis) = Beret — flat, spread out. Data is evenly distributed with no strong concentration at the centre or tails.

💬 Common Misconception: Kurtosis is often described as measuring "peakedness." While partially true, it more accurately measures tail heaviness — how much data resides in the tails versus the shoulders of the distribution. A distribution can have a sharp peak but light tails (low kurtosis) or a moderate peak but very heavy tails (high kurtosis).

Measures of Shape

Unit III · Lecture 22

🎬 Visual: Three Types of Kurtosis

Click each card to explore the shape, formula value, and real-world examples:

🏔️

Leptokurtic

β₂ > 3, γ₂ > 0

⛰️

Mesokurtic

β₂ = 3, γ₂ = 0

🏜️

Platykurtic

β₂ < 3, γ₂ < 0

Measures of Shape

Unit III · Lecture 22

📐 Karl Pearson's β₂ and Excess Kurtosis γ₂

📖 Coefficient of Kurtosis (β₂)

β₂ = μ₄ / μ₂²

Where:
• μ₂ = second central moment (variance)
• μ₄ = fourth central moment

Reference: For a normal distribution, β₂ = 3.

📖 Excess Kurtosis (γ₂)

γ₂ = β₂ − 3

The excess kurtosis measures how much the kurtosis exceeds that of a normal distribution.

  • γ₂ = 0 → Normal (mesokurtic)
  • γ₂ > 0 → Heavy tails (leptokurtic)
  • γ₂ < 0 → Light tails (platykurtic)

📖 Why μ₄ Captures Kurtosis

The fourth power heavily amplifies deviations from the mean. Values far from the mean (in the tails) contribute enormously to μ₄, while values near the mean contribute very little. Therefore:

Heavy tails → many extreme values → large μ₄ → high β₂
Light tails → few extreme values → small μ₄ → low β₂

Dividing by μ₂² standardises the measure, making it independent of scale.

Measures of Shape

Unit III · Lecture 22

🔍 Interpreting Kurtosis Values

Type	β₂	γ₂	Shape	Characteristics
Leptokurtic	> 3	> 0	Sharp peak, heavy tails	More data at centre AND tails; fewer values in the "shoulders." Higher probability of extreme values (outliers).
Mesokurtic	= 3	= 0	Normal-like shape	Similar to normal distribution. Moderate peak, moderate tails. The benchmark for comparison.
Platykurtic	< 3	< 0	Flat peak, light tails	Data is spread more uniformly. Fewer extreme values. Flatter top with shorter, thinner tails.

💡 Extreme Cases

Uniform distribution: β₂ = 1.8 (very platykurtic — perfectly flat)

Normal distribution: β₂ = 3.0 (mesokurtic — the reference)

Laplace (double exponential): β₂ = 6.0 (leptokurtic — sharp peak)

Student's t (df=4): β₂ = 9.0 (very leptokurtic — extremely heavy tails)

💡 Practical Meaning

In finance: Leptokurtic returns mean more extreme gains and losses than a normal model predicts. This is critical for risk management.

In quality control: Leptokurtic distributions suggest more defective items (far from spec) despite a good average.

In education: Platykurtic exam scores suggest uniform performance — no concentration at any particular level.

Measures of Shape

Unit III · Lecture 22

✏️ Example 1: Computing Kurtosis from Moments

📝 Problem

For a distribution, the central moments are: μ₂ = 20, μ₄ = 2500. Find β₂, γ₂, and interpret.

📖 Solution

Given: μ₂ = 20, μ₄ = 2500

β₂ = μ₄ / μ₂²
β₂ = 2500 / (20)² = 2500 / 400 = 6.25

γ₂ = β₂ − 3
γ₂ = 6.25 − 3 = +3.25

Result & Interpretation

β₂ = 6.25 | γ₂ = +3.25

Leptokurtic

The distribution is leptokurtic (heavy-tailed). β₂ = 6.25 is significantly above 3, indicating a sharper peak and heavier tails than a normal distribution. There is a higher probability of extreme values (outliers) compared to a normal distribution with the same variance.

Measures of Shape

Unit III · Lecture 22

✏️ Example 2: Kurtosis from Raw Moments

📝 Problem

The raw moments about the origin are: μ₁' = 4, μ₂' = 20, μ₃' = 100, μ₄' = 600. Find the kurtosis of the distribution.

📖 Solution — Step 1: Central Moments

μ₂ = μ₂' − (μ₁')² = 20 − 16 = 4
μ₃ = μ₃' − 3μ₂'μ₁' + 2(μ₁')³ = 100 − 3(20)(4) + 2(64) = 100 − 240 + 128 = −12
μ₄ = μ₄' − 4μ₃'μ₁' + 6μ₂'(μ₁')² − 3(μ₁')⁴
= 600 − 4(100)(4) + 6(20)(16) − 3(256)
= 600 − 1600 + 1920 − 768 = 152

📖 Solution — Step 2: Kurtosis

β₂ = μ₄ / μ₂² = 152 / 16 = 9.5

γ₂ = 9.5 − 3 = +6.5

Bonus — Skewness:
γ₁ = μ₃ / μ₂^(3/2) = −12 / 8 = −1.5 (left-skewed)

Results

β₂ = 9.5 | γ₂ = +6.5 | γ₁ = −1.5

The distribution is highly leptokurtic (very heavy tails) and strongly left-skewed. This suggests data with extreme low values and a sharp concentration near the centre-right.

Measures of Shape

Unit III · Lecture 22

✏️ Example 3: Kurtosis of Frequency Distribution

📝 Problem

Find the kurtosis for the following data:

xᵢ	2	4	6	8	10
fᵢ	3	7	10	7	3

📖 Solution

N = 30, Σfᵢxᵢ = 180, x̄ = 6
Σfᵢxᵢ² = 1200, Σfᵢxᵢ³ = 8640, Σfᵢxᵢ⁴ = 66240

μ₂' = 1200/30 = 40, μ₃' = 8640/30 = 288, μ₄' = 66240/30 = 2208

μ₂ = 40 − 36 = 4
μ₃ = 288 − 3(40)(6) + 2(216) = 288 − 720 + 432 = 0 (symmetric!)
μ₄ = 2208 − 4(288)(6) + 6(40)(36) − 3(1296) = 2208 − 6912 + 8640 − 3888 = 48

β₂ = 48 / 16 = 3.0
γ₂ = 3.0 − 3 = 0.0

Result

β₂ = 3.0 | γ₂ = 0.0 | γ₁ = 0.0

Mesokurtic & Symmetric

The distribution is mesokurtic (normal-like kurtosis) and perfectly symmetric. The symmetric frequencies (3, 7, 10, 7, 3) produce a shape very close to a normal distribution.

Measures of Shape

Unit III · Lecture 22

📏 Percentile-Based Measure of Kurtosis

📖 Definition

An alternative, robust measure of kurtosis uses percentiles (specifically quartiles and percentiles) instead of moments. This avoids the heavy influence of extreme outliers on μ₄.

📖 Percentile Kurtosis Formula (Moors' Method)

A common percentile-based kurtosis measure is:

K_perc = (P₇₅ − P₂₅) / (P₉₀ − P₁₀)
This ratio compares the interquartile range (middle 50%) to the inter-decile range (middle 80%).

Interpretation:
  • Higher K_perc → more data in the middle relative to tails → leptokurtic
  • Lower K_perc → data more evenly spread → platykurtic
  • For a normal distribution, K_perc ≈ 0.26

💡 Advantage: Percentile-based kurtosis is resistant to outliers because it does not involve raising deviations to the fourth power. It is preferred when data contains extreme values that would distort moment-based measures.

Measures of Shape

Unit III · Lecture 22

🎬 Interactive: Kurtosis Calculator

Enter your data to compute β₂, γ₂, and determine the kurtosis type:

Data values (comma-separated):

Measures of Shape

Unit III · Lecture 22

💼 Application: Kurtosis in Finance

📖 Why Kurtosis Matters for Investors

Financial returns are often assumed to follow a normal distribution. However, real market returns are typically leptokurtic — they have fatter tails than the normal distribution predicts. This means:

⚠️ The Danger of Ignoring Kurtosis

If you assume returns are normal (β₂ = 3) but they are actually leptokurtic (β₂ = 7):

• You underestimate the probability of extreme events (crashes, booms)
• Your risk models (VaR, portfolio theory) give false confidence
• "Black swan" events (rare, extreme) happen far more often than predicted

Example: The 2008 financial crisis involved daily moves that were supposed to happen "once in a billion years" under a normal model — but happened multiple times in a single month.

💡 Practical Use

1. Risk Management: High kurtosis → use fat-tailed models (Student's t, Cauchy) instead of normal.

2. Portfolio Construction: Avoid assets with very high kurtosis if you are risk-averse.

3. Option Pricing: Kurtosis affects the pricing of out-of-the-money options.

4. Stress Testing: Simulate extreme scenarios based on actual tail behaviour, not normal assumptions.

Measures of Shape

Unit III · Lecture 22

⚖️ Kurtosis vs Skewness — What Each Tells You

Feature	Skewness (γ₁)	Kurtosis (β₂ / γ₂)
What it measures	Asymmetry (direction of lean)	Tail heaviness / peakedness
Based on	Third central moment (μ₃)	Fourth central moment (μ₄)
Normal distribution	γ₁ = 0 (symmetric)	β₂ = 3, γ₂ = 0 (mesokurtic)
Key question	"Which tail is longer?"	"How heavy are the tails?"
Sign matters?	Yes (positive or negative)	β₂ is always positive; γ₂ can be negative
Independence	A distribution can be symmetric but have any kurtosis	A distribution can have any skewness but normal kurtosis
Together they tell you	Skewness + Kurtosis give a near-complete picture of a distribution's shape beyond its mean and variance.

💡 Key Insight: Skewness and kurtosis are independent characteristics. A distribution can be symmetric with heavy tails (e.g., Student's t), or skewed with normal-like tails. You need both to fully describe the shape of a distribution.

Measures of Shape

Unit III · Lecture 22

🔑 Properties of Kurtosis

β₂ ≥ 1: The minimum value of β₂ is 1 (achieved by a Bernoulli distribution with p = 0.5). There is no upper bound.
Normal distribution benchmark: β₂ = 3 (γ₂ = 0) for a normal distribution. This is the reference point for classifying kurtosis.
Kurtosis is unit-free: Since μ₄ and μ₂² have the same units (fourth power of data units), their ratio is dimensionless.
Kurtosis is invariant under linear transformation: If y = ax + b (a ≠ 0), then β₂(y) = β₂(x). Kurtosis is unaffected by changes in origin and scale.
Fourth moment amplifies extremes: Because of the fourth power, a single extreme outlier can dramatically increase μ₄ and hence β₂. This makes kurtosis sensitive to outliers.
Kurtosis measures tail weight, not peak height: While often described as "peakedness," kurtosis more precisely measures how much probability mass is in the tails. A sharp peak with thin tails can have low kurtosis.
For grouped data: Apply Sheppard's correction to μ₂ and μ₄ before computing β₂ if the data is continuous with equal class widths.
Sample kurtosis uses biased estimators: The sample β₂ is a biased estimator of the population kurtosis. For small samples, corrections exist but are beyond introductory statistics.

Measures of Shape

Unit III · Lecture 22

✏️ Example 4: Comparing Kurtosis of Two Datasets

📝 Problem

Two datasets have the following central moments. Compare their shapes.

📖 Dataset A

μ₂ = 10, μ₃ = 0, μ₄ = 300

β₂ = 300/100 = 3.0
γ₂ = 3.0 − 3 = 0.0
γ₁ = 0/… = 0

→ Mesokurtic, Symmetric

📖 Dataset B

μ₂ = 10, μ₃ = 0, μ₄ = 800

β₂ = 800/100 = 8.0
γ₂ = 8.0 − 3 = +5.0
γ₁ = 0/… = 0

→ Highly Leptokurtic, Symmetric

Comparison

Both datasets have the same variance (μ₂ = 10) and are symmetric (γ₁ = 0).

But Dataset B has much heavier tails (β₂ = 8 vs 3).

Dataset B has more extreme values (far from the mean) despite having the same spread (variance). This is the additional information that kurtosis provides beyond variance.

Measures of Shape

Unit III · Lecture 22

🔑 Summary: Measures of Kurtosis

Kurtosis measures the heaviness of tails (and peakedness) of a distribution relative to a normal distribution.
β₂ = μ₄ / μ₂² — Karl Pearson's coefficient of kurtosis. Normal distribution has β₂ = 3.
γ₂ = β₂ − 3 — Excess kurtosis. γ₂ = 0 for normal, > 0 for leptokurtic, < 0 for platykurtic.
Leptokurtic (β₂ > 3): Sharp peak, heavy tails. More extreme values. Examples: financial returns, Laplace distribution.
Mesokurtic (β₂ = 3): Normal-like shape. Moderate peak and tails. The benchmark.
Platykurtic (β₂ < 3): Flat peak, light tails. Fewer extreme values. Examples: uniform distribution.
Percentile-based kurtosis = (P₇₅ − P₂₅)/(P₉₀ − P₁₀) — a robust alternative that is resistant to outliers.
Kurtosis is invariant under linear transformations (y = ax + b). It is unit-free and independent of skewness.
Fourth moment amplifies extremes: A single outlier can dramatically increase β₂. This sensitivity is both a feature (detecting tail risk) and a limitation (susceptibility to noise).
Applications: Finance (risk modelling), quality control (detecting outlier-prone processes), meteorology (extreme weather events), insurance (catastrophe modelling).

Measures of Shape

Unit III · Lecture 22

🧠 Quick Quiz — Lecture 22

Q1: A distribution with β₂ = 5 is classified as:

A) Leptokurtic

B) Mesokurtic

C) Platykurtic

D) Symmetric

Q2: The excess kurtosis (γ₂) of a normal distribution is:

A) 3

B) 0

C) 1

D) −3

Q3: Kurtosis primarily measures:

A) The centre of the distribution

B) The asymmetry of the distribution

C) The heaviness of tails relative to a normal distribution

D) The spread of the distribution

Q4: If μ₂ = 9 and μ₄ = 1458, then β₂ equals:

A) 9

B) 18

C) 3

D) 162

Q5: A uniform distribution is:

A) Leptokurtic

B) Mesokurtic

C) Platykurtic

D) Highly skewed

Q6: If every value in a dataset is multiplied by 4, the kurtosis (β₂) will:

A) Increase 4 times

B) Increase 16 times

C) Remain the same

D) Decrease to 1/4th

Q7: Which of the following is true about kurtosis and skewness?

A) They always have the same sign

B) High kurtosis implies high skewness

C) They are independent — a distribution can have any combination of skewness and kurtosis

D) Kurtosis can only be computed if skewness is known

Q8: In finance, leptokurtic returns imply:

A) Returns are always positive

B) Extreme gains and losses occur more frequently than a normal model predicts

C) The distribution is symmetric

D) Risk is lower than expected

Measures of Shape

Unit III · Lecture 22

📝 Homework & Practice

📝 Practice Tasks

1. Given μ₂ = 25 and μ₄ = 5000, find β₂ and γ₂. Classify the distribution.

2. The raw moments of a distribution are: μ₁' = 6, μ₂' = 45, μ₃' = 360, μ₄' = 3200. Find the central moments and compute β₂, γ₂, and γ₁. Describe the shape completely.

3. For a frequency distribution with xᵢ = 1, 2, 3, 4, 5 and fᵢ = 2, 5, 8, 5, 2, compute the kurtosis. Is it mesokurtic?

4. Explain why a distribution can have β₂ > 3 but γ₁ = 0. Give a real-world example.

5. Compare the kurtosis of a normal distribution (β₂ = 3) with a Student's t-distribution with 4 degrees of freedom (β₂ = 9). What does this mean for risk assessment?

6. If the kurtosis of a grouped dataset (class width c = 10) is computed as β₂ = 4.5 before Sheppard's correction, and the uncorrected moments are μ₂ = 50 and μ₄ = 11250, find the corrected kurtosis.

7. A dataset has β₂ = 1.8. What type of distribution is this? Name a theoretical distribution that has approximately this kurtosis.

8. BONUS: Prove that the kurtosis of a Bernoulli distribution with parameter p is: β₂ = (1 − 3p(1−p)) / (p(1−p)). Evaluate this for p = 0.5 and verify that β₂ = 1 (the minimum possible kurtosis).

💡 Tip: Remember the three benchmarks: β₂ = 1.8 (uniform/platykurtic), β₂ = 3 (normal/mesokurtic), β₂ > 3 (leptokurtic). These three reference points will help you quickly classify any distribution.

Measures of Shape

Unit III · Lecture 22

Thank You!

Questions? Let's discuss!

Upcoming in Unit III

Lecture 23: Lorenz Curve & Measures of Inequality
Lecture 24: Unit III Review & Problem-Solving Session
Unit IV: Correlation & Regression Analysis

Unit III · Lecture 22 · BDS113 Introduction to Statistics · B.Sc. Data Science, Semester II