1. What Is Kurtosis?

Kurtosis is a statistical measure that describes the shape of a probability distribution, specifically how heavy-tailed or light-tailed the distribution is compared to a normal distribution. It measures the "tailedness" rather than the peakedness of the distribution.

2. Kurtosis For Normal Distribution

For a normal distribution (also called Gaussian distribution), the kurtosis value is always exactly 3. This is known as mesokurtic kurtosis.

Key Points:

• Normal distribution has kurtosis = 3 (mesokurtic)
• This serves as the benchmark for comparing other distributions
• The value 3 represents the expected fourth standardized moment

3. How To Calculate Kurtosis

The formula for calculating kurtosis from a dataset is:

Where:

• \( n \) — Number of data points
• \( x_i \) — Individual data points
• \( \bar{x} \) — Mean of the dataset
• Numerator — Fourth moment about the mean
• Denominator — Square of the variance

4. Types Of Kurtosis

Mesokurtic (Kurtosis = 3): Normal distribution, moderate tails
Leptokurtic (Kurtosis > 3): Heavy tails, more outliers
Platykurtic (Kurtosis < 3): Light tails, fewer outliers

5. Frequently Asked Questions (FAQ)

Q1: Why is kurtosis important for normal distribution?
A: Kurtosis = 3 confirms that a distribution is normal, which is fundamental for many statistical tests and assumptions.

Q2: What does excess kurtosis mean?
A: Excess kurtosis = kurtosis - 3. Positive excess kurtosis indicates heavier tails than normal, negative indicates lighter tails.

Q3: Can kurtosis be negative?
A: Yes, kurtosis values range from 1 to infinity. Values less than 3 indicate platykurtic distributions.

Q4: How is kurtosis different from skewness?
A: Skewness measures asymmetry, while kurtosis measures tail heaviness. Both describe distribution shape but different aspects.

Q5: What sample size is needed for reliable kurtosis calculation?
A: At least 20-30 data points are recommended for stable kurtosis estimates, though 4 is the mathematical minimum.