Karl Pearson’s and Bowley’s Coefficient of Skewness

Karl Pearson’s and Bowley’s Coefficient of Skewness

Get the full solved assignment PDF of MCO-022 of 2024-25 session now by clicking on above button.

Karl Pearson’s Coefficient of Skewness:

1. Definition and Formula: Karl Pearson’s Coefficient of Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It is calculated using:
   [
   \text{Skewness} = \frac{3 (\bar{x} – \text{Median})}{s}
   ]
   Where ( \bar{x} ) is the mean of the dataset, Median is the median of the dataset, and ( s ) is the standard deviation.
2. Interpretation:

• Positive Skewness: When the skewness value is positive, the distribution has a longer or fatter tail on the right side. This indicates that the bulk of the data is concentrated on the left, with a few larger values stretching out the right tail.
• Negative Skewness: When the skewness value is negative, the distribution has a longer or fatter tail on the left side. This means the bulk of the data is concentrated on the right, with a few smaller values stretching out the left tail.
• Zero Skewness: A skewness value close to zero indicates a symmetric distribution.

1. Applications: This measure is used in descriptive statistics to understand the shape of the data distribution and is useful for analyzing the skewness of various datasets in fields like finance, economics, and social sciences.

Bowley’s Coefficient of Skewness:

1. Definition and Formula: Bowley’s Coefficient of Skewness is another measure of skewness that is based on quartiles rather than the mean and median. It is calculated using:
   [
   \text{Skewness} = \frac{Q_1 + Q_3 – 2 \times \text{Median}}{Q_3 – Q_1}
   ]
   Where ( Q_1 ) and ( Q_3 ) are the first and third quartiles, respectively.
2. Interpretation:

• Positive Skewness: When the coefficient is positive, it indicates that ( Q_3 ) is greater than ( 2 \times \text{Median} – Q_1 ), showing a distribution with a longer right tail.
• Negative Skewness: When the coefficient is negative, it indicates that ( Q_1 ) is greater than ( 2 \times \text{Median} – Q_3 ), showing a distribution with a longer left tail.
• Zero Skewness: A coefficient of zero suggests a symmetric distribution.

1. Applications: Bowley’s Coefficient of Skewness is particularly useful in cases where the mean is not available or when data is better summarized by quartiles rather than by mean and standard deviation. It is often used in exploratory data analysis.

In summary, Karl Pearson’s Coefficient of Skewness measures skewness based on the mean and median, while Bowley’s Coefficient uses quartiles, each providing different perspectives on data distribution asymmetry.