Skewness and Kurtosis Calculator

Skewness and Kurtosis Formulas:

\[ Skewness = \frac{\mu_3}{\sigma^3} \quad Kurtosis = \frac{\mu_4}{\sigma^4} \]

Third Moment (μ₃):

dimensionless

Fourth Moment (μ₄):

dimensionless

Standard Deviation (σ):

dimensionless

Skewness:

Kurtosis:

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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 compared to a normal distribution.

2. How Does the Calculator Work?

The calculator uses the standard formulas:

\[ Skewness = \frac{\mu_3}{\sigma^3} \quad Kurtosis = \frac{\mu_4}{\sigma^4} \]

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, while Kurtosis values indicate how heavy the tails are compared to a normal distribution.

3. Importance of Skewness and Kurtosis

Details: These measures are crucial in statistics for understanding distribution characteristics, testing normality assumptions, risk assessment in finance, quality control, and data analysis across various fields.

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). The results are dimensionless measures.

5. Frequently Asked Questions (FAQ)

Q1: What do different skewness values indicate?
A: Positive skewness indicates right-tailed distribution, negative indicates left-tailed, and zero indicates symmetric distribution.

Q2: How to interpret kurtosis values?
A: Kurtosis > 3 indicates heavy tails (leptokurtic), < 3 indicates light tails (platykurtic), and = 3 indicates normal tails (mesokurtic).

Q3: What are typical ranges for skewness and kurtosis?
A: For normal distributions, skewness ≈ 0 and kurtosis ≈ 3. Extreme values may indicate outliers or non-normal distributions.

Q4: When are these measures most useful?
A: In financial risk analysis, quality control, scientific research, and any field requiring distribution shape analysis.

Q5: Are there limitations to these measures?
A: They can be sensitive to outliers and may not fully capture complex distribution shapes. Visual inspection is often recommended alongside these measures.