Measures of Skewness & Kurtosis: Types, Formulas & Interpretation

Key Concepts

Skewness and kurtosis describe the shape of a frequency distribution beyond its central tendency (mean, median, mode). They help businesses assess:

• Income inequality (e.g., wealth distribution in Nepal).
• Risk in investments (e.g., stock market volatility).
• Quality control (e.g., product defects clustering in manufacturing).

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1. Skewness: Measuring Symmetry

Skewness quantifies how much a distribution deviates from symmetry. A symmetric distribution has zero skewness; positive skewness means a longer right tail, and negative skewness means a longer left tail.

Types of Skewness

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Methods to Measure Skewness

(A) Quartile Coefficient of Skewness (QCS)

Formula: $$\text{QCS}=\frac{Q_3-Q_1}{Q_3+Q_1}$$

• Interpretation:
  • QCS = 0: Symmetric.
  • QCS > 0: Positive skewness.
  • QCS < 0: Negative skewness.

Worked Example 1: Daraz Order Delays Problem: A Daraz warehouse’s delivery times (in hours) have:

• Q₁ = 2 hours, Q₃ = 5 hours. Find: Quartile coefficient of skewness and interpret.

Solution: $$\text{QCS}=\frac{5-2}{5+2}=\frac{3}{7}\approx 0.43$$ Interpretation: Positive skewness (0.43) means most deliveries are fast, but a few take much longer (e.g., rural areas).

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(B) Pearson’s Coefficient of Skewness (PCS)

Formula: $$\text{PCS}=\frac{\text{Mean}-\text{Mode}}{\text{Standard Deviation}}$$

• Interpretation:
  • PCS = 0: Symmetric.
  • PCS > 0: Positive skewness.
  • PCS < 0: Negative skewness.

Worked Example 2: Ncell Data Charges Problem: For Ncell’s monthly data usage (in GB):

• Mean = 15 GB, Mode = 12 GB, SD = 4 GB. Find: PCS and interpret.

Solution: $$\text{PCS}=\frac{15-12}{4}=0.75$$ Interpretation: High positive skewness (0.75) suggests most users consume little data, but a few binge-stream heavily.

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(C) Percentile Coefficient of Skewness

Formula: $$\text{PCS}=\frac{P_{90}-P_{10}}{P_{90}+P_{10}}$$ Use case: When quartiles are unavailable but percentiles are given.

Worked Example 3: NEPSE Stock Prices Problem: NEPSE’s stock prices (Rs.) have:

• P₁₀ = 10, P₉₀ = 63. Find: Percentile coefficient of skewness.

Solution: $$\text{PCS}=\frac{63-10}{63+10}=\frac{53}{73}\approx 0.73$$ Interpretation: Strong positive skewness—most stocks are stable, but a few surge unpredictably.

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Comparison of Skewness Measures

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2. Kurtosis: Measuring Tailedness

Kurtosis measures the peakedness (sharpness) and heaviness of tails of a distribution compared to a normal curve.

Types of Kurtosis

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Measuring Kurtosis

(A) Pearson’s Coefficient of Kurtosis

Formula: $$\text{PCK}=\frac{3\times (\text{Mean}-\text{Median}{)}^2}{{\text{Standard Deviation}}^2}$$

• Interpretation:
  • PCK = 0: Mesokurtic (normal).
  • PCK > 0: Leptokurtic (peaked, heavy tails).
  • PCK < 0: Platykurtic (flat, light tails).

Worked Example 4: Pathao Ride Fares Problem: Pathao’s daily ride fares (Rs.) have:

• Mean = 200, Median = 190, SD = 15. Find: Kurtosis and interpret.

Solution: $$\text{PCK}=\frac{3\times (200-190{)}^2}{15^2}=\frac{3\times 100}{225}\approx 1.33$$ Interpretation: Leptokurtic (1.33) means most fares are similar, but a few rides are much costlier (e.g., long-distance trips).

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(B) Percentile Coefficient of Kurtosis

Formula: $$\text{PCK}=\frac{P_{90}-P_{10}}{2\times (P_{75}-P_{25})}-3$$ Use case: When quartiles/percentiles are given but mean/median are not.

Worked Example 5: NTC Call Charges Problem: NTC’s monthly call charges (Rs.) have:

• P₁₀ = 50, P₉₀ = 200, Q₁ = 70, Q₃ = 120. Find: Percentile coefficient of kurtosis.

Solution:

1. Calculate $P_{90}-P_{10}=200-50=150$.
2. Calculate $P_{75}-P_{25}=Q_3-Q_1=120-70=50$.
3. Plug into formula: $$\text{PCK}=\frac{150}{2\times 50}-3=1.5$$ Interpretation: Leptokurtic (1.5) suggests most users pay moderately, but a few have extreme charges (e.g., international calls).

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3. Relationship Between Skewness and Kurtosis

• Skewness focuses on asymmetry (tail direction).
• Kurtosis focuses on peakedness and tail weight.
• Combined interpretation:
  • A leptokurtic distribution with positive skewness (e.g., income) has few high earners and a sharp peak.
  • A platykurtic distribution with negative skewness (e.g., product defects) has few extreme failures and a flat peak.

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In the Real World

1. eSewa Transaction Volumes

   • Idea: Positive skewness in transaction amounts (most users pay small amounts, but a few make large transfers).
   • Impact: eSewa uses quartile analysis to detect fraud (e.g., sudden large payments).

2. Daraz Order Deliveries

   • Idea: Leptokurtic kurtosis in delivery times (most orders arrive on time, but a few are delayed by days).
   • Impact: Daraz optimizes last-mile logistics to reduce tail delays.

3. NEPSE Stock Returns

   • Idea: Kurtosis > 0 in stock returns (few extreme gains/losses, but high volatility).
   • Impact: Investors use kurtosis to assess risk (e.g., avoid "fat-tailed" stocks).

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Exam Tips

1. Memorize Formulas:

   • Quartile skewness: $(Q_3-Q_1)/(Q_3+Q_1)$.
   • Pearson’s skewness: $(\text{Mean}-\text{Mode})/\text{SD}$.
   • Pearson’s kurtosis: $3(\text{Mean}-\text{Median}{)}^2/{\text{SD}}^2$.

2. Interpretation Tricks:

   • Skewness > 0: Right tail longer → "Few high values."
   • Kurtosis > 0: Peaked → "Few extreme outliers."
   • Always compare to normal distribution (e.g., "more leptokurtic than normal").

3. Worked Example Strategy:

   • Step 1: Identify given data (quartiles, percentiles, mean/mode/SD).
   • Step 2: Plug into the correct formula.
   • Step 3: Interpret using business context (e.g., "income inequality," "risk").

4. Common Pitfalls:

   • Mixing skewness and kurtosis: They measure different things (symmetry vs. peakedness).
   • Ignoring units: Always state units (e.g., "QCS = 0.43 (dimensionless)").
   • Forgetting interpretation: Exams always ask for meaning (e.g., "Why is skewness positive?").

5. Time Management:

   • Skewness questions: 8–10 minutes.
   • Kurtosis questions: 10–12 minutes.
   • Interpretation: 3–5 minutes (use real-world examples like Ncell data or Daraz deliveries).

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Practice Problems (Like Exam Questions)

1. Quartile Skewness: Given $Q_1=15$, $Q_3=35$, calculate QCS and interpret for Ncell’s data usage.

2. Pearson’s Skewness: Given Mean = 50, Mode = 45, SD = 5, find PCS and discuss Pathao’s fare distribution.

3. Kurtosis: Given Mean = 100, Median = 95, SD = 10, calculate PCK and relate to NEPSE’s stock volatility.

4. Combined Interpretation: A dataset has:

   • QCS = 0.5 (positive skewness),
   • PCK = 1.2 (leptokurtic). Describe the income distribution of a Nepali city.

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Final Checklist Before Exam

• Can I calculate QCS, PCS, and PCK?
• Do I know how to interpret skewness/kurtosis?
• Can I relate these to real businesses (eSewa, Daraz, NEPSE)?
• Did I practice all past exam questions on this unit?