Kurtosis And Skewness Calculator

Skewness and Kurtosis Formulas:

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

Skewness:

Kurtosis:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is 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 moment-based formulas:

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

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: These formulas standardize the moments by the standard deviation raised to the appropriate power, making them dimensionless measures that can be compared 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) of your distribution. Standard deviation must be greater than zero. All values should be in consistent units.

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 of the distribution.

Q2: What is the kurtosis of a normal distribution?
A: A normal distribution has a kurtosis of 3. Some statistical packages subtract 3 to create "excess kurtosis" where 0 represents a normal distribution.

Q3: When are skewness and kurtosis most useful?
A: They are particularly valuable in finance for risk assessment, in quality control for process monitoring, and in research for checking normality assumptions.

Q4: Can skewness and kurtosis be negative?
A: Skewness can be negative (left-skewed), positive (right-skewed), or zero. Kurtosis is always positive since it involves even powers of deviations.

Q5: What are the limitations of these measures?
A: They can be sensitive to outliers and may not fully capture distribution shape in multimodal distributions. Large sample sizes are recommended for reliable estimates.