Skewness is a statistical measure that quantifies the asymmetry of the probability distribution of a real-valued random variable about its mean. Skewness can signify the direction and magnitude of deviation from the symmetrical bell curve or normal distribution. In mathematical terms, skewness compares the tail lengths of the distribution by computing the third standardized moment about the mean.
Mathematical Definition of Skewness
- The skewness \( \gamma_1 \) of a distribution is given by:
$$ \gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3} $$where:
- \( E \) is the expected value operator
- \( X \) is a random variable
- \( \mu \) is the mean of \( X \)
- \( \sigma \) is the standard deviation of \( X \)
Alternatively, for a sample of \( n \) values \( X_1, X_2, \ldots, X_n \), the sample skewness \( g_1 \) is:
- \( \bar{X} \) is the sample mean
- \( s \) is the sample standard deviation
Types of Skewness
Positive Skewness
- Definition: A distribution is positively skewed (or right-skewed) when the right tail is longer or fatter than the left.
- Implications: Indicates that the mean and median are greater than the mode.
- Example: Income distribution in many economies exhibits positive skewness.
Negative Skewness
- Definition: A distribution is negatively skewed (or left-skewed) when the left tail is longer or fatter than the right.
- Implications: Indicates that the mean and median are less than the mode.
- Example: The age at retirement can exhibit negative skewness.
Special Considerations
Zero Skewness
- A distribution with a skewness of zero is perfectly symmetrical, such as the standard normal distribution.
- Most real-world data are not perfectly symmetrical.
Sensitivity to Outliers
- Skewness is sensitive to outliers. A single extreme value can heavily influence the skewness calculation.
Applicability
In Finance
- Skewness helps in assessing investment returns. Positively skewed returns imply potential high returns but with some risk.
In Real Estate
- Understanding skewness helps in making pricing strategies by determining whether high prices are anomalies or a common trait.
Historical Context
- Karl Pearson introduced the concept and the coefficient of skewness in the 19th century as part of broader attempts to describe distribution shapes.
Comparisons
Skewness vs. Kurtosis
- Kurtosis: Measures the “tailedness” of the distribution.
- Skewness: Measures the asymmetry.
- Comparison: While both metrics describe distribution shape, they provide different insights into data’s frequency distribution.
Related Terms
- Kurtosis: Another moment-based statistical measure describing the tails’ extremity.
- Normal Distribution: A symmetrical distribution characterized by a skewness of zero.
- Box Plot: A graphical representation that can help identify skewness with its asymmetrical quartiles.
FAQs
Q1: Can Skewness Be Greater Than 1?
- Yes, skewness values can be greater than 1 (or less than -1) indicating strong asymmetry.
Q2: Is It Possible for Two Distributions to Have the Same Mean and Variance but Different Skewness?
- Absolutely, distributions can share the same mean and variance but differ in skewness, which affects their shape and tails.
References
- Pearson, K., “On a General Theory of Skew Correlation and Nonlinear Regression,” Drapers’ Co. Research Memoirs, Biometric Series I, 1905.
- Doane, D. P., & Seward, L. E. (2011). “Measuring Skewness: A Forgotten Statistic?” Journal of Statistics Education, 19(2).
Summary
Skewness is a vital statistical concept utilized to describe the asymmetry of data distributions. It provides critical insights in various fields such as finance, economics, and real estate by indicating the direction and the extent to which a set of values deviate from the symmetrical norm. Understanding skewness aids in comprehensively analyzing and interpreting data trends and patterns.
