Kurtosis Overview
Kurtosis measures the “tailedness” and peakedness of a distribution compared to a normal distribution. It indicates how much the distribution deviates from the normal distribution in terms of extreme values (tails).
Interpretation:
- Excess Kurtosis = 0: Mesokurtic (similar to normal distribution)
- Excess Kurtosis > 0: Leptokurtic (heavier tails, more peaked)
- Excess Kurtosis < 0: Platykurtic (lighter tails, flatter)
Note: Most statistical software uses “excess kurtosis” (Fisher’s definition), which subtracts 3 from the raw kurtosis.
Moment Coefficient of Kurtosis
Kurtosis is calculated using the fourth central moment.
Formula
For ungrouped data:
$$\beta_2 = \frac{m_4}{s^4}$$
Excess Kurtosis (Fisher’s definition):
$$\gamma_2 = \beta_2 - 3 = \frac{m_4}{s^4} - 3$$
where:
- $m_4 = \frac{\sum(x_i - \bar{x})^4}{n}$ (fourth central moment)
- $s$ = standard deviation
For grouped data:
$$\beta_2 = \frac{\sum f_i(m_i - \bar{x})^4}{n \cdot s^4}$$
Kurtosis for Ungrouped Data
Example: Calculate Kurtosis
Test scores of 8 students: 45, 50, 55, 60, 65, 70, 75, 80
Calculate the moment coefficient of kurtosis.
Solution:
Step 1: Calculate mean
$$\bar{x} = \frac{45 + 50 + 55 + 60 + 65 + 70 + 75 + 80}{8} = \frac{500}{8} = 62.5$$
Step 2: Create deviation table
| $x_i$ | $(x_i - \bar{x})$ | $(x_i - \bar{x})^2$ | $(x_i - \bar{x})^4$ |
|---|---|---|---|
| 45 | -17.5 | 306.25 | 93,789.06 |
| 50 | -12.5 | 156.25 | 24,414.06 |
| 55 | -7.5 | 56.25 | 3,164.06 |
| 60 | -2.5 | 6.25 | 39.06 |
| 65 | 2.5 | 6.25 | 39.06 |
| 70 | 7.5 | 56.25 | 3,164.06 |
| 75 | 12.5 | 156.25 | 24,414.06 |
| 80 | 17.5 | 306.25 | 93,789.06 |
| 1050 | 242,812.5 |
Step 3: Calculate variance and standard deviation
$$s^2 = \frac{1050}{8} = 131.25$$
$$s = \sqrt{131.25} = 11.456$$
$$s^4 = (11.456)^4 = 17,216.8$$
Step 4: Calculate fourth central moment
$$m_4 = \frac{242,812.5}{8} = 30,351.56$$
Step 5: Calculate kurtosis
$$\beta_2 = \frac{30,351.56}{17,216.8} = 1.763$$
Excess Kurtosis:
$$\gamma_2 = 1.763 - 3 = -1.237$$
Interpretation: Platykurtic (flatter than normal, lighter tails)
Kurtosis for Grouped Data
Example: Grouped Data
| Class | Midpoint | Frequency |
|---|---|---|
| 10-20 | 15 | 4 |
| 20-30 | 25 | 8 |
| 30-40 | 35 | 12 |
| 40-50 | 45 | 10 |
| 50-60 | 55 | 6 |
Calculate kurtosis (assume mean = 34.5, s = 11.8).
Solution:
| Class | $m_i$ | $f$ | $(m_i - \bar{x})$ | $(m_i - \bar{x})^4$ | $f(m_i - \bar{x})^4$ |
|---|---|---|---|---|---|
| 10-20 | 15 | 4 | -19.5 | 144,850 | 579,400 |
| 20-30 | 25 | 8 | -9.5 | 8,145 | 65,160 |
| 30-40 | 35 | 12 | 0.5 | 0.06 | 0.72 |
| 40-50 | 45 | 10 | 10.5 | 121,551 | 1,215,510 |
| 50-60 | 55 | 6 | 20.5 | 176,661 | 1,059,966 |
| N=40 | 2,920,037 |
$$m_4 = \frac{2,920,037}{40} = 73,000.9$$
$$s^4 = (11.8)^4 = 19,356.5$$
$$\beta_2 = \frac{73,000.9}{19,356.5} = 3.77$$
$$\gamma_2 = 3.77 - 3 = 0.77$$
Interpretation: Leptokurtic (heavier tails, more peaked than normal)
Types of Kurtosis
Leptokurtic (Excess Kurtosis > 0)
- More peaked than normal distribution
- Heavier tails (more extreme values)
- Sharper peak in center
- Interpretation: Data has more outliers than normal distribution
Mesokurtic (Excess Kurtosis ≈ 0)
- Similar to normal distribution
- Moderate peak and tails
- Interpretation: Distribution matches normal distribution
Platykurtic (Excess Kurtosis < 0)
- Flatter than normal distribution
- Lighter tails (fewer extreme values)
- Lower, broader peak
- Interpretation: Data is more uniformly distributed
Visual Representation
Leptokurtic Mesokurtic Platykurtic
(Heavy tails) (Normal) (Light tails)
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Excess > 0 Excess ≈ 0 Excess < 0
Relationship to Other Measures
Combined with Skewness:
- Normal Distribution: Skewness ≈ 0, Excess Kurtosis ≈ 0
- Skewed with Heavy Tails: Non-zero skewness, Excess Kurtosis > 0
- Symmetric, Flat: Skewness ≈ 0, Excess Kurtosis < 0
Interpretation Guidelines
| Excess Kurtosis | Type | Characteristics |
|---|---|---|
| < -1 | Very Platykurtic | Very flat, rare outliers |
| -1 to 0 | Platykurtic | Flatter than normal |
| 0 | Mesokurtic | Similar to normal |
| 0 to 1 | Leptokurtic | Peaked, moderate outliers |
| > 1 | Very Leptokurtic | Very peaked, many outliers |
Kurtosis in Different Distributions
| Distribution | Excess Kurtosis |
|---|---|
| Normal | 0 |
| Uniform | -1.2 |
| Exponential | 6.0 |
| Student’s t (5 df) | 6.0 |
Applications
- Financial Analysis: Detecting market crashes (heavy tails)
- Risk Management: Identifying extreme value risk
- Quality Control: Understanding process variability
- Data Distribution: Assessing normality
- Model Selection: Choosing appropriate statistical methods
Important Notes
- Kurtosis is sensitive to extreme values
- For small samples, kurtosis estimates can be unstable
- Excess kurtosis (Fisher’s definition) is more commonly used in practice
- Always interpret kurtosis alongside skewness for complete picture
- Kurtosis interpretation assumes large sample sizes (n > 30)
Limitations
- Highly sensitive to extreme values and outliers
- Requires larger sample sizes for reliable estimation
- Different formulas may give different interpretations
- Should not be used alone without considering other distribution characteristics