Skewness and Kurtosis Formulas
Skewness measures the asymmetry of a distribution; kurtosis measures the heaviness of the tails.
Both are essential for understanding non-normal data.
The Formulas
Sample skewness: g&sub1; = [n / ((n−1)(n−2))] × Σ((xi−x̄)/s)³
Kurtosis: γ&sub2; = E[(X−μ)&sup4;] / σ&sup4;
Excess kurtosis = γ&sub2; − 3
Skewness measures the asymmetry of a probability distribution around its mean. Kurtosis measures how heavy the tails are (how prone to extreme values). Together, they describe the shape of a distribution beyond just the mean and standard deviation.
Variables and Interpretation
| Symbol | Meaning |
|---|---|
| γ&sub1; = 0 | Symmetric distribution (e.g., normal distribution) |
| γ&sub1; > 0 | Positive (right) skew — tail extends further to the right; mean > median |
| γ&sub1; < 0 | Negative (left) skew — tail extends further to the left; mean < median |
| Excess kurtosis = 0 | Mesokurtic — normal distribution tails |
| Excess kurtosis > 0 | Leptokurtic — heavier tails than normal (more outliers) |
| Excess kurtosis < 0 | Platykurtic — lighter tails than normal (fewer outliers) |
Example — Dataset Calculation
Dataset: {2, 4, 4, 5, 5, 7, 9}. Calculate skewness.
n = 7, mean x̄ = (2+4+4+5+5+7+9)/7 = 36/7 = 5.143
Deviations: −3.143, −1.143, −1.143, −0.143, −0.143, 1.857, 3.857
s² = Σ(x−x̄)²/(n−1) = (9.878 + 1.306 + 1.306 + 0.020 + 0.020 + 3.449 + 14.877)/6 = 30.857/6 = 5.143
s = 2.268
Σ((x−x̄)/s)³ = Σ of cubed standardized deviations
Skewness g&sub1; ≈ +0.56 — slight positive skew; the value 9 pulls the tail to the right
When to Use Them
Use skewness and kurtosis when:
- Testing whether data follows a normal distribution before applying parametric tests
- Analyzing financial returns — stock returns typically have negative skew and excess kurtosis (fat tails)
- Quality control — skewed process data may indicate systematic problems
- Income distribution analysis — income data is almost always right-skewed
- Risk management — heavy-tailed distributions mean extreme events are more common than normal models predict
Rules of thumb for normality: skewness between −0.5 and +0.5 is roughly symmetric; between ±0.5 and ±1.0 is moderately skewed; beyond ±1.0 is highly skewed. Excess kurtosis beyond ±2 suggests significant deviation from normality.