Calculate Skewness Using Variance and Third Moment
Professional Statistical Distribution Analyzer
Formula: √Variance
Formula: (σ²) ^ 1.5
γ₁ = μ₃ / σ³
Visual Distribution Representation
Figure: Dynamic curve showing left vs. right asymmetry based on current inputs.
What is Skewness Calculated Using Variance and Third Moment?
To calculate skewness using variance and third moment is a fundamental process in descriptive statistics used to determine the asymmetry of a probability distribution. While the mean and variance tell us about the center and the spread of data, skewness informs us about the shape—specifically, whether the data leans more heavily toward one tail than the other.
In many financial and scientific models, assuming a perfectly normal (symmetric) distribution can lead to errors. By choosing to calculate skewness using variance and third moment, analysts can quantify “tail risk.” A positive skew indicates a long right tail, while a negative skew indicates a long left tail. This is critical for risk management and standardizing datasets across different scales.
The Formula and Mathematical Explanation
The mathematical relationship required to calculate skewness using variance and third moment relies on the concept of moments. Specifically, skewness is the third standardized moment.
The formula is expressed as:
Where:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| γ₁ (Gamma 1) | Fisher’s Skewness Coefficient | Dimensionless | -3.0 to +3.0 (common) |
| μ₃ (Mu 3) | Third Central Moment | Units³ | Any Real Number |
| σ² (Sigma Squared) | Variance | Units² | Positive Real Number |
| σ | Standard Deviation | Units | Positive Real Number |
Practical Examples
Example 1: Analyzing Investment Returns
Suppose an investment portfolio has a variance of 0.04 (standard deviation of 20%) and a third central moment of 0.002. To calculate skewness using variance and third moment:
- Variance (σ²) = 0.04
- Third Moment (μ₃) = 0.002
- Standard Deviation (σ) = √0.04 = 0.2
- σ³ = 0.2 * 0.2 * 0.2 = 0.008
- Skewness = 0.002 / 0.008 = 0.25
Interpretation: The distribution is slightly positively skewed, meaning there is a higher probability of small losses but a potential for occasional large gains.
Example 2: Manufacturing Quality Control
A factory measures the diameter of bolts. The variance is 9 mm², and the third moment is -54 mm³. To calculate skewness using variance and third moment:
- Variance = 9
- Third Moment = -54
- σ = 3, so σ³ = 27
- Skewness = -54 / 27 = -2.0
Interpretation: Significant negative skew, indicating most bolts are slightly larger than the mean, with a few outliers being much smaller.
How to Use This Skewness Calculator
- Enter the Variance: Input the second central moment of your dataset. Note that this must be a positive value.
- Enter the Third Moment: Input the third central moment. This value can be positive, negative, or zero.
- Read the Result: The calculator will instantly calculate skewness using variance and third moment and display the coefficient.
- Check the Visualization: The dynamic SVG chart will update to show you a visual representation of the asymmetry.
- Copy Data: Use the “Copy Result Details” button to save your findings for a report or spreadsheet.
Key Factors That Affect Skewness Results
- Outliers: Since the third moment cubes the deviations from the mean, outliers have a massive impact on the skewness coefficient.
- Sample Size: In small samples, the estimated third moment can be highly volatile, leading to unstable skewness results.
- Data Range: Large variances generally spread the denominator, which can “dampen” the skewness if the third moment doesn’t grow proportionally.
- Distribution Type: Log-normal distributions are naturally positively skewed, while power-law distributions exhibit extreme skewness.
- Data Cleaning: Removing extreme values often drastically reduces the skewness calculated using variance and third moment.
- Aggregation Level: Aggregating data (e.g., daily to monthly) often normalizes the distribution, reducing skewness toward zero.
Frequently Asked Questions (FAQ)
Yes, skewness values are not bounded between -1 and 1. Highly skewed data can have coefficients of 3, 5, or even higher.
If you calculate skewness using variance and third moment and the third moment is zero, the skewness is zero, indicating a perfectly symmetric distribution.
Variance is the average of squared deviations. Squaring any real number results in a non-negative value. If variance were negative, the square root (standard deviation) would be an imaginary number.
No. While a Normal Distribution has zero skewness, other symmetric distributions (like a Uniform or Laplace distribution) also have zero skewness.
Skewness measures asymmetry (third moment), while kurtosis measures “tailedness” or peakiness (fourth moment). Both are essential for statistical distribution analysis.
Yes, you first calculate the mean, then the variance (average squared deviation), and then the third moment (average cubed deviation) before using this tool.
Negative skew (left-skewed) means the left tail is longer or fatter than the right tail. The mass of the distribution is concentrated on the right.
This “standardizes” the moment, making skewness a dimensionless number that allows comparison between different datasets regardless of their units.
Related Tools and Internal Resources
- Variance Calculator: Calculate the second central moment from raw data.
- Standard Deviation Calculator: Find the square root of variance for dispersion analysis.
- Kurtosis Calculator: Explore the fourth moment to understand tail weight.
- Probability Density Function Tool: Visualize how skewness affects the PDF curve.
- Moment Generating Function Guide: Deep dive into the calculus behind statistical moments.
- Data Normality Testing: Use skewness and kurtosis to test for normality.