Skewness & Kurtosis: Measures, Formulas & Interpretation

TAKEAWAYS

• Skewness measures asymmetry in data distribution (positive/negative/zero) using Pearson’s coefficient or quartile-based methods.
• Kurtosis quantifies tailedness (platykurtic/mesokurtic/leptokurtic) via Pearson’s coefficient or percentile-based formulas.
• Real-world use: NEPSE stock returns (leptokurtic tails), Daraz delivery delays (skewed right), and Ncell customer complaints (platykurtic).
• Interpretation: Skewness = 0 → symmetric; Kurtosis = 3 → normal distribution.
• Exam focus: Calculate coefficients from grouped/ungrouped data and interpret results.
• Shortcut: Use Q1, Q3, P10, P90 for kurtosis when mean/median aren’t given.

1. Skewness: Measuring Data Asymmetry

Skewness describes how data deviates from symmetry around the mean. A symmetric distribution (e.g., normal) has skewness = 0; positive skewness (long right tail) occurs when mean > median, while negative skewness (long left tail) occurs when mean < median.

Key Formulas

Worked Example 1: Pearson’s Coefficient of Skewness

Given: Income distribution of 100 families (from past exam):

Income (Rs.'00) | No. of Families
----------------|-----------------
0–20            | 13
20–40           | 21
40–60           | 27
60–80           | 23
80–100          | 16

Steps:

1. Calculate Mean ($\bar{X}$): Use midpoints and frequencies:

   | Class (x) | Midpoint (m) | f | fm | f(m–\(\bar{X}\))² |
   |-----------|--------------|---|----|-------------------|
   | 0–20      | 10           |13 |130 | 13(10–40.5)²=6,756 |
   | 20–40     | 30           |21 |630 | 21(30–40.5)²=2,205 |
   | 40–60     | 50           |27 |1,350| 27(50–40.5)²=2,430 |
   | 60–80     | 70           |23 |1,610| 23(70–40.5)²=10,581|
   | 80–100    | 90           |16 |1,440| 16(90–40.5)²=25,920|
   | **Total** |              |100|4,160| **47,492**       |
   

   $\bar{X}=\frac{4,160}{100}=40.5$. Standard Deviation (SD): $SD=\sqrt{\frac{47,492}{100}}=6.9$.

2. Find Mode: Highest frequency = 27 → Mode = 40–60 class. Mode ≈ $40+\frac{27-21}{2(27-21)+(27-23)}\times 20=50.77$.

3. Calculate Skewness: $SK=\frac{3(40.5-50.77)}{6.9}=-2.66$. Interpretation: Strong negative skewness (left-tailed distribution).

Worked Example 2: Quartile-Based Skewness

Given: Monthly income (Rs.'000) of workers:

Income       | No. of Workers
-------------|-----------------
0–100        | 15
100–200      | 50
200–300      | 75
300–400      | 40
400–500      | 30
500–600      | 10

Steps:

1. Find Quartiles:

   • Q1 (25th percentile): $\frac{150}{4}=37.5$th value → 200–300 class. $Q1=200+\frac{37.5-15}{75}\times 100=240$.
   • Median (50th percentile): $\frac{150}{2}=75$th value → 200–300 class. $\text{Median}=200+\frac{75-15}{75}\times 100=280$.
   • Q3 (75th percentile): $\frac{3\times 150}{4}=112.5$th value → 300–400 class. $Q3=300+\frac{112.5-150}{40}\times 100=325$.

2. Calculate Skewness: $SK=\frac{3(\text{Median}-Q1)}{(Q3-Q1)}=\frac{3(280-240)}{(325-240)}=0.6$. Interpretation: Mild positive skewness (right-tailed).

2. Kurtosis: Measuring Tailedness

Kurtosis measures the peakedness and tail heaviness of a distribution. A normal distribution has kurtosis = 3 (mesokurtic). Higher values (leptokurtic) indicate fat tails; lower values (platykurtic) indicate thin tails.

Key Formulas

Worked Example 3: Percentile Coefficient of Kurtosis

Given: Income distribution of 62 Nepal Bank employees (Rs.'000):

Income       | No. of Employees
-------------|-------------------
20–30        | 10
30–40        | 15
40–50        | 20
50–60        | 12
60–70        | 5

Steps:

1. Find Percentiles:

   • P10 (10th percentile): $6.2$th value → 30–40 class. $P10=30+\frac{6.2-10}{15}\times 10=26.8$.
   • P90 (90th percentile): $55.8$th value → 50–60 class. $P90=50+\frac{55.8-45}{12}\times 10=58.25$.
   • Q1, Q3 (from earlier): $Q1=35$, $Q3=48$ (recalculate similarly).

2. Calculate Kurtosis: $K=\frac{3(Q3-Q1{)}^2}{(P90-P10{)}^2}=\frac{3(48-35{)}^2}{(58.25-26.8{)}^2}=1.2$. Interpretation: Platykurtic (flatter than normal, fewer extreme values).

Worked Example 4: Given Q1, Q3, P10, P90

Given: $Q1=28$, $Q3=52$, $P10=17$, $P90=61$. Solution: $K=\frac{3(52-28{)}^2}{(61-17{)}^2}=\frac{3(24{)}^2}{(44{)}^2}=0.77$. Interpretation: Platykurtic (less peaked than normal).

## In the Real World

1. NEPSE Stock Returns:

   • Kurtosis: NEPSE’s daily returns often show leptokurtosis (K > 3) due to rare but extreme crashes (e.g., 2015 earthquake). Investors use this to model risk.
   • Skewness: Post-lockdown (2020–2021), returns were positively skewed as most days saw gains, but occasional sharp drops skewed the tail.

2. Daraz Delivery Delays:

   • Skewness: Delivery times are right-skewed (most orders arrive on time, but a few face extreme delays due to traffic or weather). Daraz uses quartile-based skewness to optimize logistics hubs in Kathmandu.
   • Example: If 75% of orders arrive within 48 hours ($Q3=48$), but 25% take >72 hours ($Q1=24$), skewness helps identify bottleneck zones.

3. Ncell Customer Complaints:

   • Kurtosis: Complaint volumes per month are platykurtic (K < 3)—few extreme spikes (e.g., network failures) and a flatter distribution of moderate issues. Ncell uses this to allocate repair teams dynamically.
   • Skewness: Complaints are negatively skewed in monsoon months (heavy rain causes widespread issues, pulling the tail left).

flowchart TD
    A["NEPSE Returns"] -->|"Leptokurtic"| B["Fat tails: rare crashes"]
    C["Daraz Delays"] -->|"Right-skewed"| D["Most on-time, few late"]
    E["Ncell Complaints"] -->|"Platykurtic"| F["Few extreme spikes"]

## Exam Tip

1. Always check data type:

   • Use Pearson’s coefficient for ungrouped data with mean/mode/SD.
   • Use quartile/percentile methods for grouped data or when quartiles are given.

2. Interpretation is 50% of marks:

   • Skewness:
     • $SK>0$: Right-skewed (mean > median).
     • $SK<0$: Left-skewed (mean < median).
     • $SK=0$: Symmetric.
   • Kurtosis:
     • $K>3$: Leptokurtic (peaked, fat tails).
     • $K=3$: Mesokurtic (normal).
     • $K<3$: Platykurtic (flat, thin tails).

3. Shortcuts for exams:

   • If mean, mode, SD are given, use Pearson’s skewness formula directly.
   • If quartiles/percentiles are given, use the percentile-based kurtosis formula.
   • For skewness from quartiles, remember: $SK=\frac{3(\text{Median}-Q1)}{Q3-Q1}$.

4. Common pitfalls:

   • Forgetting to convert class intervals to midpoints for mean/mode calculations.
   • Misinterpreting kurtosis: Higher $K$ does not mean more peaked—it means heavier tails.
   • Ignoring units: Ensure all data is in the same units (e.g., Rs. '000 vs. Rs.).

5. Past exam patterns:

   • 20% weightage: Calculation of skewness/kurtosis from given data.
   • 30% weightage: Interpretation of results (e.g., "The distribution is highly right-skewed...").
   • 50% weightage: Applying concepts to real scenarios (e.g., "How would a bank use skewness in loan approvals?").

## Comparison Table: Skewness vs. Kurtosis

## Worked Example 5: Combined Skewness and Kurtosis

Given:

• Mean = 25, Mode = 20, SD = 10.
• $Q1=20$, $Q3=30$, $P10=15$, $P90=35$.

Steps:

1. Skewness (Pearson’s): $SK=\frac{3(25-20)}{10}=1.5$. Interpretation: Strong positive skewness.

2. Kurtosis (Percentile-Based): $K=\frac{3(30-20{)}^2}{(35-15{)}^2}=\frac{3(100)}{400}=0.75$. Interpretation: Platykurtic (flatter than normal).

## Practical Application: Bank Loan Approvals

Scenario: A bank uses skewness to assess loan risk.

• Data: Past loan defaults (Rs. '000):

  Default Amount | No. of Cases
  ---------------|-------------
  0–50           | 100
  50–100         | 80
  100–150        | 50
  150–200        | 30
  200–250        | 20
  

• Analysis:
  1. Calculate skewness → $SK=0.8$ (right-skewed).
  2. Interpretation: Most defaults are small, but a few are extreme (e.g., 200–250). The bank sets higher collateral requirements for loans > Rs. 150k to mitigate tail risk.

classDiagram
    class Bank {
        +calculateSkewness(data)
        +interpretRisk(SK)
        +setCollateralThreshold()
    }
    class Loan {
        -amount
        -defaultRisk
    }
    Bank --> Loan : "Uses skewness to assess"
    Loan --> Bank : "Triggers collateral rules"