Moment Coefficient of Kurtosis Calculator
Use this unified calculator to find the moment coefficient of kurtosis for both ungrouped (raw) data and grouped (frequency distribution) data. Measure how peaked or flat a distribution is compared to the normal distribution.
Quick Start
Choose your data type, enter your values, and click Calculate:
| Moment Coefficient of Kurtosis Calculator | |
|---|---|
| Data Type | Ungrouped (Raw Data) Grouped (Frequency Distribution) |
| Enter the X Values (Separated by comma,) | |
| Type of Frequency Distribution | DiscreteContinuous |
| Enter the Classes for X (Separated by comma,) | |
| Enter the frequencies (f) (Separated by comma,) | |
| Results | |
| Number of Observations (N): | |
| Mean of X values: | |
| First Central Moment (μ₁): | |
| Second Central Moment (μ₂): | |
| Third Central Moment (μ₃): | |
| Fourth Central Moment (μ₄): | |
| Coefficient of Kurtosis (β₂): | |
| Coefficient of Kurtosis (γ₂): | |
Understanding Kurtosis
Kurtosis (from Greek “kurtos” = curved) measures the degree of peakedness and tail heaviness of a distribution:
- ✅ Compares shape to normal distribution
- ✅ Measures tail behavior (propensity for outliers)
- ✅ Captures distribution peakedness
- ✅ Two forms: β₂ (algebraic) and γ₂ (excess, practical)
Three Distribution Types
| Type | β₂ | γ₂ | Shape | Tails |
|---|---|---|---|---|
| Leptokurtic | > 3 | > 0 | Peaked/Sharp | Heavy |
| Mesokurtic | = 3 | = 0 | Normal | Moderate |
| Platykurtic | < 3 | < 0 | Flat | Light |
Formulas
Coefficient of Kurtosis (β₂ Form)
$$\beta_2 = \frac{\mu_4}{\mu_2^2}$$
Where:
- $\mu_4$ = Fourth central moment
- $\mu_2$ = Second central moment (variance)
Coefficient of Kurtosis (γ₂ Form) - Excess Kurtosis
$$\gamma_2 = \beta_2 - 3$$
Or directly:
$$\gamma_2 = \frac{\mu_4}{\mu_2^2} - 3$$
Why Subtract 3?
The normal distribution has β₂ = 3, so:
- γ₂ = 0: Distribution has normal kurtosis (mesokurtic)
- γ₂ > 0: More peaked than normal (leptokurtic)
- γ₂ < 0: Flatter than normal (platykurtic)
How to Use This Calculator
For Ungrouped (Raw) Data
Step 1: Select “Ungrouped (Raw Data)”
Step 2: Enter your data values separated by commas
Step 3: Click “Calculate”
Results will show:
- Mean and all central moments
- Both kurtosis coefficients (β₂ and γ₂)
- Distribution classification
For Grouped (Frequency Distribution) Data
Step 1: Select “Grouped (Frequency Distribution)”
Step 2: Choose distribution type and enter data
Step 3: Click “Calculate”
Distribution Types Explained
Leptokurtic (Peaked)
- β₂ > 3 or γ₂ > 0
- Sharp peak, pronounced central concentration
- Heavy tails (more outliers)
- Examples: Financial returns, stock prices (frequent extremes)
Mesokurtic (Normal)
- β₂ = 3 or γ₂ = 0
- Moderate peak, typical bell curve
- Normal tail behavior
- Examples: Many natural phenomena, test scores
Platykurtic (Flat)
- β₂ < 3 or γ₂ < 0
- Flat distribution, spread out
- Light tails (fewer outliers)
- Examples: Uniform distributions, rainfall distribution
Worked Examples
Example 1: Ungrouped Data - Normal Distribution
Data: Values: 3, 5, 7, 9, 11, 13, 15
Find the coefficient of kurtosis.
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{3+5+7+9+11+13+15}{7} = 9$$
Step 2: Calculate Central Moments
Deviations: -6, -4, -2, 0, 2, 4, 6
μ₁ = 0 μ₂ = (36+16+4+0+4+16+36)/7 = 112/7 = 16 μ₃ = (-216-64-8+0+8+64+216)/7 = 0/7 = 0 μ₄ = (1296+256+16+0+16+256+1296)/7 = 3136/7 = 448
Step 3: Calculate Kurtosis
$$\beta_2 = \frac{448}{16^2} = \frac{448}{256} = 1.75$$
$$\gamma_2 = 1.75 - 3 = -1.25$$
Answer: β₂ = 1.75, γ₂ = -1.25
Interpretation: The distribution is platykurtic (flatter than normal) with γ₂ < 0. This uniform distribution has a flat shape and light tails.
Example 2: Ungrouped Data - Peaked Distribution
Data: Values: 5, 5, 5, 5, 5, 10, 20
Find the coefficient of kurtosis.
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{5+5+5+5+5+10+20}{7} = 8.57$$
Step 2: Calculate Central Moments
Deviations: -3.57, -3.57, -3.57, -3.57, -3.57, 1.43, 11.43
μ₂ = (12.74+12.74+12.74+12.74+12.74+2.04+130.64)/7 = 43.29 μ₃ = (-45.53-45.53-45.53-45.53-45.53+2.92+1493.08)/7 = 122.34 μ₄ = (162.5+162.5+162.5+162.5+162.5+4.18+17045.5)/7 = 2661.16
Step 3: Calculate Kurtosis
$$\beta_2 = \frac{2661.16}{43.29^2} = \frac{2661.16}{1874}{} = 1.42$$
Actually let me recalculate: β₂ = 2661.16/(43.29)² = 2661.16/1874.22 = 1.42
$$\gamma_2 = 1.42 - 3 = -1.58$$
Answer: β₂ ≈ 1.42, γ₂ ≈ -1.58
Interpretation: Despite concentration, this distribution is still platykurtic due to extreme values causing high variance.
Example 3: Grouped Data (Discrete) - Test Scores
Problem: Score distribution for 50 students. Calculate kurtosis.
| Score | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| Frequency | 5 | 10 | 20 | 10 | 5 |
Solution:
Step 1: Calculate Mean
$$\overline{x} = \frac{3(5)+4(10)+5(20)+6(10)+7(5)}{50} = \frac{250}{50} = 5$$
Step 2: Calculate Central Moments
Deviations: -2, -1, 0, 1, 2
μ₂ = [5(4)+10(1)+20(0)+10(1)+5(4)]/50 = 60/50 = 1.2 μ₃ = [5(-8)+10(-1)+20(0)+10(1)+5(8)]/50 = 0/50 = 0 μ₄ = [5(16)+10(1)+20(0)+10(1)+5(16)]/50 = 180/50 = 3.6
Step 3: Calculate Kurtosis
$$\beta_2 = \frac{3.6}{1.2^2} = \frac{3.6}{1.44} = 2.5$$
$$\gamma_2 = 2.5 - 3 = -0.5$$
Answer: β₂ = 2.5, γ₂ = -0.5
Interpretation: The distribution is platykurtic (flatter than normal). Scores are distributed symmetrically with a flatter shape than a normal distribution.
Example 4: Grouped Data (Continuous) - Age Distribution
Problem: Age distribution of 60 people. Calculate kurtosis.
| Age Group | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|---|---|---|---|---|---|
| Frequency | 3 | 8 | 35 | 10 | 4 |
Solution:
Step 1: Use Midpoints and Calculate Mean
| Midpoint | Frequency | f×x |
|---|---|---|
| 25 | 3 | 75 |
| 35 | 8 | 280 |
| 45 | 35 | 1,575 |
| 55 | 10 | 550 |
| 65 | 4 | 260 |
| Total | 60 | 2,740 |
$$\overline{x} = 2740/60 = 45.67$$
Step 2: Calculate Central Moments
Deviations: -20.67, -10.67, -0.67, 9.33, 19.33
μ₂ = [3(427.7)+8(113.9)+35(0.45)+10(87.0)+4(373.6)]/60 = 3584/60 = 59.73
μ₄ = [3(183.5k)+8(13k)+35(0.2)+10(7.6k)+4(139k)]/60 = 1,052,000/60 = 17,533
Step 3: Calculate Kurtosis
$$\beta_2 = \frac{17533}{59.73^2} = \frac{17533}{3567.7} = 4.91$$
$$\gamma_2 = 4.91 - 3 = 1.91$$
Answer: β₂ ≈ 4.91, γ₂ ≈ 1.91
Interpretation: Highly leptokurtic distribution (γ₂ > 1). The sharp concentration of ages around 40-50 creates a peaked distribution with heavy tails relative to normal.
Relationship Between Skewness and Kurtosis
| Skewness | Kurtosis | Distribution Shape |
|---|---|---|
| 0 | 0 | Normal (Bell curve) |
| 0 | > 0 | Peaked, symmetric |
| 0 | < 0 | Flat, symmetric |
| ≠ 0 | 0 | Skewed, normal tails |
| ≠ 0 | ≠ 0 | Both skewed and unusual peak |
Kurtosis Applications
Financial Markets
- High kurtosis: Stock returns show frequent extreme movements
- Management: Risk must account for tail behavior
- Trading: Adjust strategies for outlier probability
Quality Control
- High kurtosis: Process produces occasional defects
- Low kurtosis: Process very stable and consistent
Medical Research
- Normal kurtosis: Drug response follows expected pattern
- High kurtosis: Some patients have extreme reactions
Common Mistakes to Avoid
❌ WRONG: Confusing β₂ and γ₂ ✓ RIGHT: β₂ uses normal as 3; γ₂ uses normal as 0 (excess kurtosis)
❌ WRONG: Thinking high kurtosis always means “bad” ✓ RIGHT: Context matters;financial risk vs. quality control trade-offs
❌ WRONG: Ignoring skewness when interpreting kurtosis ✓ RIGHT: Consider both skewness and kurtosis together
❌ WRONG: Using only outliers to judge kurtosis ✓ RIGHT: Kurtosis is about all deviations, not just extremes
Key Differences: Ungrouped vs. Grouped Data
| Aspect | Ungrouped | Grouped |
|---|---|---|
| Precision | Exact moments | Approximate from midpoints |
| Calculation | Direct from values | Using class midpoints |
| Data Loss | None | Some from grouping |
| Computational | More values | Simplified with frequencies |
Visual Interpretation
Leptokurtic (Peaked, Heavy Tails)
- Sharp central peak
- Tails don’t drop off as quickly
- More mass in extremes
- Better for extreme value analysis
Mesokurtic (Normal, Moderate Tails)
- Bell curve shape
- Standard tail behavior
- Reference standard
- Common in nature
Platykurtic (Flat, Light Tails)
- Spread out, flat top
- Tails drop off quickly
- Few extreme values
- Uniform-like behavior
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