1. What Is Kurtosis?

Kurtosis is a statistical measure that describes the shape of a probability distribution, specifically the "tailedness" and peakedness compared to a normal distribution. It helps identify whether data are heavy-tailed or light-tailed relative to a normal distribution.

2. How Does The Calculator Work?

The calculator uses the population kurtosis formula:

Where:

• \( x_i \) — Individual data points
• \( \mu \) — Mean of the data
• \( \sigma \) — Standard deviation of the data
• \( n \) — Number of data points

Example Calculation: For a normal distribution, kurtosis = 3. For a uniform distribution, kurtosis ≈ 1.8.

3. Types Of Kurtosis

Mesokurtic: Kurtosis = 3 (normal distribution)
Leptokurtic: Kurtosis > 3 (heavy tails, more peaked)
Platykurtic: Kurtosis < 3 (light tails, less peaked)
Excess Kurtosis: Kurtosis - 3 (often used to compare to normal distribution)

4. Using The Calculator

Instructions: Enter your data values separated by commas. The calculator will compute the mean, standard deviation, and kurtosis. Ensure you have at least 4 data points for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What does kurtosis tell us about a distribution?
A: Kurtosis measures the tail heaviness and peak sharpness. High kurtosis indicates more outliers, while low kurtosis suggests fewer outliers.

Q2: Why is normal distribution kurtosis = 3?
A: This is the mathematical result of the kurtosis formula applied to a normal distribution. Many statistical packages report excess kurtosis (kurtosis - 3).

Q3: What's the difference between skewness and kurtosis?
A: Skewness measures asymmetry, while kurtosis measures tail behavior and peakedness.

Q4: When is high kurtosis problematic?
A: High kurtosis in financial data may indicate higher risk of extreme events. In quality control, it may suggest process instability.

Q5: Can kurtosis be negative?
A: Yes, platykurtic distributions have kurtosis less than 3, which can be negative when using excess kurtosis.