Coefficient Of Skewness And Kurtosis Formula

Moment Coefficients Formulas:

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

Coefficient of Skewness:

Coefficient of Kurtosis:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 Does the Calculator Work?

The calculator uses the moment coefficient 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 coefficients are dimensionless measures that allow comparison of distribution shapes across different datasets and scales.

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, fourth moment, and standard deviation. All values must be valid (standard deviation > 0). The results are dimensionless coefficients.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the difference between moment kurtosis and excess kurtosis?
A: Moment kurtosis is calculated as shown above. Excess kurtosis subtracts 3 to compare against the normal distribution (excess kurtosis = kurtosis - 3).

Q3: What are typical ranges for skewness and kurtosis?
A: For normal distribution: skewness ≈ 0, kurtosis ≈ 3. Significant deviations from these values indicate non-normal distributions.

Q4: When are these measures most useful?
A: In statistical analysis, quality control, financial modeling, and any field where understanding distribution shape is important for making inferences.

Q5: Can these measures be misleading?
A: Yes, with small sample sizes or when outliers heavily influence the moments. Always consider sample size and data quality when interpreting results.