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 standard formulas:

Where:

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

Explanation: Skewness values indicate the direction and degree of asymmetry (positive = right-skewed, negative = left-skewed). Kurtosis values indicate the heaviness of tails compared to normal distribution.

3. Importance of Skewness and Kurtosis

Details: These measures are crucial in statistics for understanding distribution characteristics, testing normality assumptions, and identifying outliers in data analysis.

4. Using the Calculator

Tips: Enter the third moment (μ₃), fourth moment (μ₄), and standard deviation (σ). All values must be valid (standard deviation > 0).

5. Frequently Asked Questions (FAQ)

Q1: What do positive and negative skewness values mean?
A: Positive skewness indicates a longer right tail (mean > median), negative skewness indicates a longer left tail (mean < median).

Q2: What are typical kurtosis values?
A: Normal distribution has kurtosis = 3. Values > 3 indicate heavier tails (leptokurtic), values < 3 indicate lighter tails (platykurtic).

Q3: When are these measures most useful?
A: Essential in financial analysis, quality control, and any field requiring distribution shape analysis and normality testing.

Q4: Are there limitations to these formulas?
A: Sensitive to outliers and sample size. Alternative formulas exist for small samples or specific distributions.

Q5: How do these relate to normal distribution?
A: For normal distribution, skewness = 0 and kurtosis = 3. Deviations indicate departure from normality.