Skewed Distribution: Definition, Visualization and Example

In the world of statistics, we often talk about distributions. A distribution, at its core, is simply a way to show how often different values occur within a dataset. While many introductory statistics lessons focus on the perfectly symmetrical and beautiful Normal Distribution (also known as the Bell Curve), the reality is that much of the data we encounter in the real world isn’t so neatly organized. That’s where understanding skewed distributions becomes crucial.

This blog post aims to provide a comprehensive understanding of skewed distributions, covering what they are, why they occur, how to identify them, and what their implications are for statistical analysis. Whether you’re a student grappling with statistical concepts, a researcher analyzing data, or simply curious about the world around you, this guide will equip you with the knowledge to navigate the nuances of skewed distributions.

What is a Skewed Distribution?

A skewed distribution, quite simply, is a distribution that is not symmetrical. In a symmetrical distribution, like the Normal Distribution, the mean, median, and mode are all equal and located at the center of the curve. The left and right sides of the distribution are mirror images of each other.

In contrast, a skewed distribution has a “tail” that stretches out further on one side than the other. This tail pulls the mean away from the median and mode, disrupting the symmetry. We classify skewed distributions based on the direction of this tail:

• Right-Skewed Distribution (Positively Skewed): The tail is longer on the right side of the distribution. This means there are more high values that are significantly larger than the majority of the data. The mean is typically greater than the median and mode.
• Left-Skewed Distribution (Negatively Skewed): The tail is longer on the left side of the distribution. This means there are more low values that are significantly smaller than the majority of the data. The mean is typically less than the median and mode.

Visualizing Skewed Distributions:

Imagine a histogram, a common visual representation of a distribution. In a right-skewed distribution, the bulk of the data is clustered towards the left side of the histogram, with a long tail extending towards the right. Conversely, in a left-skewed distribution, the bulk of the data is clustered towards the right, with a long tail extending towards the left.

Key Differences Between Symmetrical and Skewed Distributions:

Why Do Skewed Distribution Occur?

Skewed distributions are common because many real-world phenomena are naturally asymmetrical. Here are some common reasons why they occur:

• Lower Bound: Many variables have a natural lower bound of zero. For example, income, age of first employment, or number of customers visiting a store in an hour cannot be negative. If a large portion of the data is close to this lower bound, and there’s potential for much larger values, the distribution will likely be right-skewed. Think of income – many people earn relatively low incomes, but a few earn exceptionally high incomes, creating a long right tail.
• Upper Bound: Conversely, some variables have a natural upper bound. For example, test scores often have a maximum score. If many people score close to the maximum, the distribution will likely be left-skewed.
• Outliers: The presence of outliers, which are extreme values that deviate significantly from the rest of the data, can heavily influence the skewness of a distribution. A few very high outliers can pull the mean towards the right, resulting in right-skewness. Similarly, a few very low outliers can pull the mean towards the left, resulting in left-skewness.
• Data Collection Errors: While less common, skewed distributions can sometimes arise from data collection errors. For instance, if a measurement tool is biased, it might consistently underreport or overreport values, leading to a skewed distribution.
• Nature of the Phenomenon: The underlying nature of the phenomenon being studied can naturally lead to skewed distributions. For example, the distribution of lifespan for electronic devices is often right-skewed because most devices last a reasonable amount of time, but a few fail prematurely.

Examples of Skewed Distribution in Real Life

• Income Distribution (Right-Skewed): As mentioned earlier, income is a classic example of a right-skewed distribution. The vast majority of people earn a moderate income, but a small percentage earn extremely high incomes.
• Age at Death (Left-Skewed): In developed countries with good healthcare, the age at death is often left-skewed. Most people live to a relatively old age, with a smaller number dying at younger ages due to accidents, diseases, or other unforeseen circumstances.
• Exam Scores (Left-Skewed): If an exam is relatively easy, many students will score high marks, leading to a left-skewed distribution.
• Reaction Time (Right-Skewed): In psychological experiments measuring reaction time, most people respond quickly, but some experience delays, resulting in a right-skewed distribution.
• Website Traffic (Right-Skewed): The number of visits per day to a website is often right-skewed. Most days will have a typical amount of traffic, but occasionally a popular article or promotion will lead to a significant spike in visits.

How to Identify Skewed Distribution

Several methods can be used to identify skewed distributions:

• Visual Inspection:
  • Histograms: As mentioned earlier, histograms provide a clear visual representation of the distribution’s shape. Look for the long tail extending to the left or right.
  • Box Plots: Box plots display the median, quartiles, and outliers of a dataset. In a skewed distribution, the median will not be centered within the box, and the whiskers will have different lengths.
• Statistical Measures:
  • Skewness Coefficient: This is a numerical measure of asymmetry. A skewness coefficient of 0 indicates a symmetrical distribution. A positive skewness coefficient indicates right-skewness, and a negative skewness coefficient indicates left-skewness. Different formulas exist for calculating skewness, such as Pearson’s moment coefficient of skewness and adjusted Fisher-Pearson standardized moment coefficient.
  • Relationship between Mean, Median, and Mode: As previously discussed, the relationship between these measures can provide clues about skewness.

Impact of Skewness on Statistical Analysis:

Understanding skewness is crucial because it can significantly impact statistical analysis and the interpretation of results.

• Choice of Summary Statistics:
  • Mean: The mean is sensitive to outliers and skewed data. In a skewed distribution, the mean can be misleading as a measure of central tendency because it is pulled towards the tail.
  • Median: The median is less sensitive to outliers and skewed data. It is a more robust measure of central tendency for skewed distributions.
  • Mode: The mode represents the most frequent value. It can be informative, but less useful as a sole measure of central tendency in skewed distributions.
• Selection of Statistical Tests:
  • Many statistical tests, such as t-tests and ANOVA, assume that the data is normally distributed. If the data is significantly skewed, applying these tests directly can lead to inaccurate results.
  • Non-parametric Tests: For skewed data, non-parametric tests, such as the Mann-Whitney U test or the Wilcoxon signed-rank test, are often more appropriate because they do not rely on the assumption of normality.
• Data Transformation:
  • In some cases, data transformation techniques can be used to reduce skewness and make the data more suitable for parametric statistical tests.
  • Log Transformation: Log transformation is commonly used to reduce right-skewness.
  • Square Root Transformation: Square root transformation is another option for reducing right-skewness, often used when the data includes zeros.
  • Box-Cox Transformation: This is a more general transformation technique that can be used to handle a wider range of skewness.
• Interpretation of Results:
  • When interpreting statistical results, it’s important to consider the skewness of the data.
  • For example, if you are calculating confidence intervals for the mean of a right-skewed distribution, the upper bound of the interval may be significantly larger than the lower bound due to the influence of outliers.

Mitigating the Effects of Skewness

Here’s a summary of how to address the challenges posed by skewed distributions:

1. Identify Skewness: Use visual methods (histograms, box plots) and statistical measures (skewness coefficient) to determine the presence and direction of skewness.
2. Consider the Median: Use the median as a measure of central tendency rather than the mean. Report the median and interquartile range (IQR) for a more robust summary of the data.
3. Choose Appropriate Tests: If you need to perform statistical tests, consider using non-parametric tests designed for non-normal data.
4. Transform the Data: If possible, apply data transformation techniques like log transformation or Box-Cox transformation to reduce skewness and normalize the data. Remember to back-transform the results after analysis for proper interpretation.
5. Report Skewness: Always report the skewness of your data when presenting your results. This allows readers to understand the shape of the distribution and interpret the findings appropriately.

Conclusion

Skewed distributions are a common and important feature of real-world data. Understanding what they are, why they occur, and how to identify them is essential for accurate statistical analysis and interpretation. By considering the impact of skewness on your analysis and employing appropriate techniques, you can ensure that your conclusions are valid and meaningful. So, next time you encounter a distribution that doesn’t quite fit the bell curve, remember the power of understanding skewness! Data Science Blog