1. What Are Skewness and Kurtosis?

Skewness and Kurtosis are statistical measures that describe the shape of a probability distribution. Skewness measures the asymmetry of the distribution, while Kurtosis measures the "tailedness" or peakiness of the distribution.

2. How Do the Formulas Work?

The calculator uses the standardized moment formulas:

Where:

• \( \mu_3 \) — Third central moment (measure of asymmetry)
• \( \mu_4 \) — Fourth central moment (measure of tail heaviness)
• \( \sigma \) — Standard deviation of the distribution

Explanation: These formulas standardize the third and fourth moments by dividing by the standard deviation raised to the appropriate power, making them dimensionless and comparable across different distributions.

3. Importance of Skewness and Kurtosis

Details: Skewness helps identify if data is symmetric (skewness ≈ 0), right-skewed (positive), or left-skewed (negative). Kurtosis indicates whether data has heavy tails (leptokurtic, kurtosis > 3), light tails (platykurtic, kurtosis < 3), or normal tails (mesokurtic, kurtosis ≈ 3).

4. Using the Calculator

Tips: Enter the third moment (μ₃), fourth moment (μ₄), and standard deviation (σ). All values must be valid with standard deviation greater than zero. The results are dimensionless measures.

5. Frequently Asked Questions (FAQ)

Q1: What does positive skewness indicate?
A: Positive skewness means the distribution has a longer right tail, with most data points concentrated on the left side.

Q2: What is excess kurtosis?
A: Excess kurtosis is kurtosis minus 3 (the kurtosis of a normal distribution). Positive excess kurtosis indicates heavier tails than normal.

Q3: When are these measures most useful?
A: They are particularly valuable in finance for risk assessment, in quality control for process monitoring, and in research for data distribution analysis.

Q4: What are typical ranges for skewness and kurtosis?
A: For normal distributions, skewness ≈ 0 and kurtosis ≈ 3. Values beyond ±2 for skewness and significantly different from 3 for kurtosis suggest non-normal distributions.

Q5: Can these measures be misleading?
A: Yes, with small sample sizes or outliers, these measures can be unreliable. They should be interpreted alongside other descriptive statistics and visualizations.