4.6 Negative Skewness (Left Skew)

Negative skewness, also known as left skewness, describes a data distribution where the tail on the left side of the probability density function is longer or fatter than the tail on the right side. This indicates that most of the data points are concentrated towards the higher values, while a few extreme low values pull the distribution's tail to the left.

Key Characteristics of Negative Skewness

• Tail Direction: The left tail (lower values) is significantly longer or more stretched out than the right tail.
• Data Concentration: The majority of data points are clustered towards the higher end of the distribution.
• Central Tendency Relationship: The mean is typically less than the median, which is typically less than the mode.

Relationship Between Central Tendency: Mean < Median < Mode

This specific ordering of central tendency measures is a hallmark of negative skewness.

• Mean: Highly sensitive to extreme values. The few very low values in a negatively skewed distribution will pull the mean down, making it lower than most of the data points.
• Median: The middle value when the data is ordered. It is less affected by extreme outliers than the mean.
• Mode: The most frequently occurring value. In a negatively skewed distribution, the mode is often found at the peak of the distribution, representing the highest concentration of data, which is typically at the higher end.

This relationship, Mean < Median < Mode, arises because the mean is pulled towards the extreme low values, while the mode remains at the peak of the data concentration.

Why Negative Skewness Matters

Understanding negative skewness is crucial for accurate data interpretation and analysis:

• Influence on the Mean: The presence of extreme low values can significantly distort the mean, making it a misleading indicator of the typical or central value in the dataset.
• Impact on Statistical Analysis: Many statistical techniques assume data is normally distributed. Negatively skewed data can violate these assumptions, leading to inaccurate results and potentially flawed conclusions for methods like t-tests or linear regression if not properly addressed.
• Decision-Making: Outliers, especially those in the left tail, can disproportionately influence predictions, forecasts, and strategic decisions based on the data. For example, a few very low sales figures could dramatically reduce the average sales performance, impacting business strategies.

Examples of Negatively Skewed Distributions

• Exam Scores on an Easy Exam: If an exam is very easy, most students will score high marks (e.g., 80-100%). A few students who perform exceptionally poorly (e.g., scoring below 30%) will create a long left tail, representing the "failed badly" group.
• Age at Retirement: Most people might retire in their late 50s or early 60s. However, a small number of individuals might retire much earlier due to health reasons, disability, or early retirement packages, forming a left tail for earlier retirement ages.
• Gestational Age at Birth: The majority of babies are born at or around full term (e.g., 38-40 weeks). Premature births, while less common, represent a significant left tail of the distribution for gestational age.
• Income in a Highly Skilled Profession: In fields like technology or medicine, a few individuals might earn exceptionally high salaries, but the majority of practitioners will earn solid, high incomes. However, if there's a segment of the profession that earns significantly less due to experience or specific roles, it can create a left skew.

Visual Summary

Detecting Negative Skewness: Karl Pearson's Skewness Coefficient

Karl Pearson's coefficient of skewness is a common method for quantifying skewness.

Using Mean and Median:

The formula using the mean and median is:

Sk = (3 * (Mean - Median)) / Standard Deviation

• If Sk < 0: The distribution is negatively skewed.
• If Sk ≈ 0: The distribution is approximately symmetrical.
• If Sk > 0: The distribution is positively skewed.

This formula is particularly useful because it relies on measures of central tendency that have different sensitivities to extreme values.

Conclusion

Recognizing and understanding negative skewness is critical for accurate data interpretation. When a dataset exhibits negative skewness, relying solely on the mean to represent the central tendency can be misleading. In such cases, using the median often provides a more robust and representative measure of the typical value. Adjusting your analytical approach, such as employing the median or considering data transformations, can lead to a more accurate understanding of the data distribution and inform better decision-making.