19.X.2.4. Skewness

Skewness = $\frac{1}{\sqrt{\lambda}}$

Skewness is a numerical measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. In the context of the Poisson distribution, skewness is expressed by the formula: $\frac{1}{\sqrt{\lambda}}$, where $\lambda$ is the average number of events. This formula indicates that the skewness of the Poisson distribution is inversely related to the square root of the mean, $\lambda$. As $\lambda$ increases, the skewness decreases, suggesting that the distribution becomes more symmetric.

Consider a factory that produces a few defective products per week, represented by a small $\lambda$. If the average reflects a low defect rate (e.g., $\lambda = 1$), the distribution will be quite skewed, indicating that most weeks have zero defects, with only a few weeks reporting one. However, if the factory improves its quality control and the average jumps to ten defects per week (e.g., $\lambda = 10$), the distribution becomes more symmetric, meaning the occurrence of defects varies more evenly across weeks.

This shows that the distribution becomes more symmetric as $\lambda$ increases.

As the mean rate of events ($\lambda$) increases, the skewness of the Poisson distribution decreases. This implies that with a higher average event rate, the probability distribution approaches a more normal shape—symmetric about the mean. A lower $\lambda$ results in a highly skewed distribution where most of the probability mass is concentrated on lower values of $k$, while higher values become less probable.

Imagine two scenarios: on a slow day at a restaurant with a low average of two customers per hour ($\lambda = 2$), there are many hours with no customers, leading to high skewness. On a busy Saturday with an average of 20 customers per hour ($\lambda = 20$), customer arrivals are more evenly spread throughout the day. The resulting distribution of customer counts becomes much more balanced and less skewed around the mean, reflecting a typical busy day pattern.

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