Asymmetrical distribution
Asymmetrical distribution is a statistical distribution where values are not evenly distributed around the central tendency. The left and right sides of the distribution curve are not mirror images. Skewness measures this asymmetry quantitatively. Management researchers encounter asymmetrical distributions frequently when analyzing salary data, company sizes, customer spending patterns, and risk metrics.
Types of asymmetry
Positive skew (right-skewed)
A positively skewed distribution has a longer right tail. Most values cluster on the left side. The mean exceeds the median in most cases. Income distributions typically exhibit positive skew. A few high earners pull the average above what most people earn.
Examples in business contexts include:
- Executive compensation at Fortune 500 companies
- Startup valuations in venture capital portfolios
- Customer lifetime values in subscription businesses
- Sales commission earnings
Negative skew (left-skewed)
Negatively skewed distributions show longer left tails. Values concentrate on the right side. The mean falls below the median typically. Student test scores in easy examinations often display negative skew. Most students perform well, while a few struggle.
Business applications include:
- Customer satisfaction scores (most satisfied, few dissatisfied)
- Equipment failure times near end of useful life
- Quality control measurements approaching upper limits
Zero skew
Symmetric distributions have zero skew. The normal distribution is the most recognized example. Height and weight in large populations approximate symmetry. Manufacturing tolerances often follow normal distributions when processes are stable.
Measurement methodology
Pearson's coefficient
Karl Pearson developed the first standardized measure of skewness in the 1890s. His first coefficient uses the mode:
Skewness = (Mean - Mode) / Standard Deviation
The second coefficient substitutes the median:
Skewness = 3(Mean - Median) / Standard Deviation
This formula was widely adopted because the median is more reliably estimated than the mode from sample data.[1]
Fisher's moment coefficient
Ronald Fisher refined skewness measurement in his 1930 work on statistical estimation. The third standardized moment provides:
g₁ = m₃ / m₂^(3/2)
Where m₂ and m₃ represent the second and third central moments. This measure became standard in statistical software packages.
Bowley's quartile coefficient
Arthur Bowley proposed a robust alternative in 1920. His coefficient uses quartiles:
SK = (Q₃ + Q₁ - 2Q₂) / (Q₃ - Q₁)
This measure resists the influence of extreme outliers. It proves useful when analyzing data with potential measurement errors.
Interpretation standards
Statisticians developed guidelines for interpreting skewness values. The conventions are:
| Skewness value | Interpretation |
|---|---|
| -0.5 to +0.5 | Approximately symmetric |
| -1.0 to -0.5 or +0.5 to +1.0 | Moderately skewed |
| Below -1.0 or above +1.0 | Highly skewed |
Bulmer's 1979 text Principles of Statistics established these thresholds. They remain standard references in applied statistics courses.
Implications for analysis
Central tendency selection
The choice of average matters greatly with asymmetrical data. The mean is pulled toward the tail. Real estate markets illustrate this principle. Median home prices better represent typical values than mean prices, which luxury properties inflate.
Financial management applications require careful consideration. Mean portfolio returns may overstate typical investor experience when return distributions are right-skewed.
Statistical testing
Many parametric tests assume normality. The t-test, ANOVA, and regression methods all have this requirement. Substantial skewness violates these assumptions. Results may be misleading.
Solutions include:
- Logarithmic transformations for right-skewed data
- Square root transformations for count data
- Non-parametric alternatives like Mann-Whitney U test
Risk management applications
Value at Risk (VaR) calculations depend on distributional assumptions. Financial returns exhibit negative skew during crises. Standard VaR models underestimate tail risk when they assume symmetry.
The 2008 financial crisis revealed these limitations. Portfolio losses exceeded VaR predictions at many institutions. J.P. Morgan's RiskMetrics group subsequently incorporated skewness and kurtosis into their models.
Software implementation
Statistical packages calculate skewness automatically. SPSS uses Fisher's coefficient by default. R provides the moments package with multiple skewness measures. Excel's SKEW function implements the adjusted Fisher-Pearson coefficient. Python's SciPy library offers comprehensive options through scipy.stats.skew().
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References
- Pearson, K. (1895). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society A, 186, 343-414.
- Fisher, R.A. (1930). The Genetical Theory of Natural Selection. Clarendon Press.
- Bulmer, M.G. (1979). Principles of Statistics. Dover Publications.
Footnotes
[1] Pearson's median-based skewness coefficient gained widespread adoption in the early 20th century as it proved more stable with limited sample sizes than mode-based alternatives.