Calculate Skewness Using Variance and Third Moment

Analyze the asymmetry of your data distribution accurately using the Pearson’s moment coefficient of skewness formula.

Standard Deviation (σ)
5.0000

Standard Deviation Cubed (σ³)
125.0000

Calculation Formula
γ₁ = μ₃ / σ³

What is Calculate Skewness Using Variance and Third Moment?

To calculate skewness using variance and third moment is to quantify the lack of symmetry in a probability distribution or dataset. In statistics, skewness measures how much a distribution deviates from the normal “bell curve.” When you calculate skewness using variance and third moment, you are specifically utilizing the Pearson’s moment coefficient of skewness, which is the most common standardized measure of asymmetry.

Who should use this method? Data scientists, financial analysts, and researchers often calculate skewness using variance and third moment to understand if their data is prone to outliers on one side of the mean. A common misconception is that skewness depends only on the mean and median; however, to precisely calculate skewness using variance and third moment, one must look at higher-order moments that capture the weight of the distribution’s tails.

Calculate Skewness Using Variance and Third Moment Formula

The mathematical approach to calculate skewness using variance and third moment involves dividing the third central moment by the cube of the standard deviation. Since the standard deviation is the square root of variance, the relationship is deeply intertwined with the spread of the data.

Variable	Meaning	Unit	Typical Range
μ₃	Third Central Moment	Units³	-∞ to +∞
σ²	Variance	Units²	0 to +∞
σ	Standard Deviation	Units	0 to +∞
γ₁	Skewness Coefficient	Dimensionless	-3 to +3 (common)

Mathematical Derivation

The step-by-step logic to calculate skewness using variance and third moment is as follows:

• Step 1: Obtain the Variance (σ²).
• Step 2: Calculate the Standard Deviation (σ) by taking the square root of the Variance.
• Step 3: Calculate the cube of the Standard Deviation (σ³).
• Step 4: Divide the Third Central Moment (μ₃) by σ³ to get the Skewness (γ₁).

Practical Examples

Example 1: Financial Return Distribution

An analyst wants to calculate skewness using variance and third moment for a stock’s monthly returns. The variance is 0.04 and the third moment is -0.002.

1. σ = √0.04 = 0.2

2. σ³ = 0.008

3. Skewness = -0.002 / 0.008 = -0.25.

Interpretation: The data is slightly negatively skewed, indicating more frequent small gains but occasional large losses.

Example 2: Quality Control in Manufacturing

To calculate skewness using variance and third moment for parts length: Variance = 9 mm², Third Moment = 54 mm³.

1. σ = √9 = 3

2. σ³ = 27

3. Skewness = 54 / 27 = 2.0.

Interpretation: Highly positive skew, suggesting most parts are near the minimum length with a few significantly longer outliers.

How to Use This Calculate Skewness Using Variance and Third Moment Calculator

Follow these instructions to get the most out of this tool:

1. Enter Variance: Input the variance of your dataset. Ensure this value is positive as variance cannot be negative.
2. Enter Third Moment: Input the third central moment (μ₃). This value can be negative, zero, or positive.
3. Review Real-time Results: The calculator will immediately calculate skewness using variance and third moment.
4. Analyze the Chart: Look at the visual curve to see how the skewness translates to a distribution shape.
5. Copy and Share: Use the copy button to save your results for reports or further analysis.

Key Factors That Affect Calculate Skewness Using Variance and Third Moment Results

• Outliers: Extreme values significantly impact the third moment, causing skewness to spike.
• Sample Size: Smaller samples may produce unreliable results when you calculate skewness using variance and third moment.
• Data Range: Broad ranges often result in higher variance, which dampens the skewness coefficient unless the third moment is also very high.
• Market Volatility: In finance, high volatility increases variance, requiring a massive third moment to show high skewness.
• Measurement Precision: Errors in data collection can distort the third moment more than the mean or variance.
• Underlying Distribution: Knowing if the data follows a specific law (like Power Law) helps interpret why you calculate skewness using variance and third moment in a certain way.

Frequently Asked Questions (FAQ)

1. Why do we cube the standard deviation?

We cube it to make the skewness coefficient “dimensionless.” Since the third moment is in units cubed, dividing by standard deviation cubed cancels out the units.

2. Can skewness be zero?

Yes. When you calculate skewness using variance and third moment and get zero, it means the distribution is perfectly symmetrical (like a Normal distribution).

3. What does a negative skewness mean?

It means the tail on the left side of the distribution is longer or fatter than the right side.

4. Is variance always required?

Yes, because skewness is a standardized moment, and standardization requires the standard deviation derived from the variance.

5. Is skewness the same as kurtosis?

No. While you calculate skewness using variance and third moment to find asymmetry, kurtosis uses the fourth moment to measure “tailedness” or peak intensity.

6. How does skewness affect financial risk?

Negative skewness implies “crash risk,” where there is a higher probability of extreme negative outcomes than a normal distribution would suggest.

7. Can I calculate skewness with only the mean?

No, the mean only tells you the center. You must calculate skewness using variance and third moment or use the raw data points.

8. What is a “highly skewed” value?

Generally, a skewness value greater than 1 or less than -1 is considered highly skewed.