Definition
An asymmetrical distribution is a probability distribution in which the two halves on either side of the central point (mean, median, or mode) are not mirror images of each other. This is commonly observed in skewed distributions, where the data points are not evenly distributed around a central value.
Types of Asymmetrical Distributions
Positive Skew (Right Skew)
In distributions with a positive skew, the tail on the right side of the distribution is longer or fatter than the left side. The mean and median are typically greater than the mode.
Negative Skew (Left Skew)
In distributions with a negative skew, the tail on the left side of the distribution is longer or fatter than the right side. The mean and median are typically less than the mode.
Properties of Asymmetrical Distributions
- Skewness: A measure of the asymmetry of the probability distribution. Positive skewness indicates a right-skewed distribution, while negative skewness indicates a left-skewed distribution.
- Kurtosis: Describes the “tailedness” of the distribution. Asymmetrical distributions may have high or low kurtosis depending on the prominence of tails.
- Central Tendency Measures: Mean, median, and mode are not equal in asymmetrical distributions. The relationship among these measures indicates the direction of skewness.
- Tail Behavior: One tail may be heavier than the other, indicating the presence of outliers or extreme values.
Examples
- Income Distribution: Typically, income distribution is right-skewed where most people have lower incomes, and a few have extremely high incomes.
- Housing Prices: Often exhibit a right skew due to the high prices of luxury properties.
- Examination Scores: Sometimes show left-skewed distributions if a significant number of students score high and fewer score low.
Historical Context
The study of skewed distributions dates back to Karl Pearson in the early 20th century, who developed measures of skewness and the Pearson family of curves to describe different types of asymmetrical distributions.
Applicability
Statistics and Data Analysis
Understanding asymmetrical distributions is crucial for accurate data interpretation, as it affects the choice of statistical methods and models.
Economics and Finance
Identifying skewed distributions helps in risk management, investment decisions, and understanding market behaviors.
Comparisons
- Symmetrical Distribution: Both sides are mirror images (e.g., Normal Distribution).
- Bimodal Distribution: Contains two peaks; can be both symmetrical and asymmetrical.
Related Terms
- Normal Distribution: A symmetrical distribution represented by a bell curve (Gaussian distribution).
- Outliers: Data points that lie far from other observations; often found in skewed distributions.
- Mode: The value that appears most frequently in a data set.
FAQs
What is the main cause of asymmetrical distribution?
How is skewness measured?
Is a skewed distribution always a bad thing?
References
- Pearson, K. (1895). “Contributions to the Mathematical Theory of Evolution.” Philosophical Transactions of the Royal Society.
- Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). “Continuous Univariate Distributions.” Wiley.
Summary
Asymmetrical distribution is a fundamental concept in statistics, characterized by unequal distribution of data points around a central value. Understanding its properties, causes, and implications is essential for accurate data analysis in various fields.
This content structure ensures our Encyclopedia entry on “Asymmetrical Distribution” is comprehensive, well-organized, and accessible for readers seeking in-depth information on the topic.
