In the study of Statistics, after understanding Measures of Central Tendency (like mean, median, and mode) and measures of dispersion (like range, variance, and standard deviation), we move towards shape-related characteristics of the distribution. Two such characteristics are Skewness and Kurtosis. While skewness deals with the asymmetry of the distribution, kurtosis addresses the Peakedness or flatness of a distribution curve. This article is dedicated to understanding Kurtosis in depth.
What is Kurtosis?
Kurtosis is a statistical measure that describes the shape of a distribution's tails in relation to its overall shape. In simpler terms, it tells us about the thickness or heaviness of the tails and the sharpness of the peak of a frequency distribution curve compared to a normal distribution.
While the normal distribution has a moderate peak and tails, kurtosis helps us understand if the data is more or less outlier-prone than the normal curve.
Types of Kurtosis
- Mesokurtic – Normal distribution, standard kurtosis (baseline for comparison)
- Leptokurtic – More peaked and heavier tails than normal
- Platykurtic – Flatter peak and lighter tails than normal
Conceptual Understanding of Kurtosis
To understand kurtosis better, visualize the height and tails of the bell-shaped curve. Kurtosis measures:
- Peakedness: How tall or short the central peak is.
- Tail weight: How heavy or light the tails are (i.e., how many outliers are present).
It is important to remember that kurtosis is not only about the peak. It is also about the outliers and tail extremity.
Formula for Kurtosis
The most commonly used formula for kurtosis is the moment-based formula. For a dataset with n values:
Kurtosis (β2) =
Where:
x = each individual value
𝜇 = mean of the data
n = number of observations
Excess Kurtosis = β2 - 3
Interpretation of Excess Kurtosis
- If β2 = 3 → Mesokurtic (normal curve)
- If β2 > 3 → Leptokurtic (heavy tails)
- If β2 < 3 → Platykurtic (light tails)
Example:
Let’s take a small dataset: 4, 5, 6, 7, 8
Mean (𝜇) = (4+5+6+7+8)/5 = 6
Deviations: -2, -1, 0, +1, +2
Squared deviations: 4, 1, 0, 1, 4 → Sum = 10
Fourth powers: 16, 1, 0, 1, 16 → Sum = 34
Variance = 10/5 = 2
Standard Deviation2 = 2
Kurtosis = (34/5) / (2)2 = 6.8 / 4 = 1.7
Since 1.7 < 3 → Platykurtic
Merits of Kurtosis
- Gives insights into the probability of extreme values (outliers).
- Helps in identifying the nature of data distribution beyond mean and variance.
- Useful in risk analysis, especially in finance and economics.
Demerits of Kurtosis
- Can be sensitive to outliers, leading to misleading results in small datasets.
- Often misunderstood as a measure of peakedness only.
- Not as intuitively clear as measures like mean or standard deviation.
Skewness vs Kurtosis
| Aspect | Skewness | Kurtosis |
|---|---|---|
| Focus | Asymmetry of distribution | Peakedness / Tail weight |
| Symmetry | Yes – measures deviation from symmetry | No – symmetrical curves can differ in kurtosis |
| Normal Value | 0 | 3 (Excess = 0) |
| Types | Positive, Negative, Zero | Leptokurtic, Mesokurtic, Platykurtic |
Applications of Kurtosis
- Finance: Risk management to assess the probability of extreme market returns.
- Quality Control: Analyzing process behavior and defects.
- Economics: Studying income distribution extremities.
- Psychology: Distribution analysis in behavioral studies.
Key Points to Remember
- Kurtosis is a fourth-moment statistic.
- It’s about the tails and outliers, not just the height of the peak.
- Use kurtosis with other measures (mean, variance, skewness) for comprehensive understanding.
Conclusion
Kurtosis is a valuable statistical tool for analyzing the nature of data distribution, especially when studying how data behaves in the extremes. It complements skewness by offering another dimension to understand the shape of a distribution. For UGC NET Commerce aspirants, mastering kurtosis is essential not just for exams but also for real-world data interpretation in finance, economics, and business analytics.