Understanding Kelly's Coefficient of Skewness

Kelly's coefficient of skewness is a measure used in statistics to describe the asymmetry of a probability distribution. Unlike Pearson's coefficient, which uses mean and standard deviation, Kelly's coefficient uses percentiles (or deciles). It is particularly useful when the data is not symmetrical or contains outliers, as percentiles are less sensitive to extreme values than the mean.

The coefficient helps us understand if the data is skewed to the left (negatively skewed), skewed to the right (positively skewed), or symmetric. A positive coefficient indicates positive skewness (tail on the right), a negative coefficient indicates negative skewness (tail on the left), and a coefficient near zero suggests symmetry.

Formula for Kelly's Coefficient of Skewness

Kelly's coefficient of skewness (Sk) is calculated using the 10th, 50th, and 90th percentiles. The formula is given by:

$$ \text{Sk} = \frac{P_{90} + P_{10} - 2P_{50}}{P_{90} - P_{10}} $$

Where:

• $P_{90}$ is the 90th percentile
• $P_{10}$ is the 10th percentile
• $P_{50}$ is the 50th percentile (which is also the median)

Given Percentile Partition Values

We are provided with the following percentile partition values for the distribution:

• $P_{10} = 58.983$
• $P_{50} = 61.345$
• $P_{90} = 63.831$

Step-by-Step Calculation of Kelly's Coefficient

Now, let's substitute the given values into Kelly's coefficient formula and calculate the skewness.

Step 1: Calculate the numerator

The numerator is $P_{90} + P_{10} - 2P_{50}$.

$$ \text{Numerator} = 63.831 + 58.983 - 2 \times 61.345 $$ $$ \text{Numerator} = 122.814 - 122.690 $$ $$ \text{Numerator} = 0.124 $$

Step 2: Calculate the denominator

The denominator is $P_{90} - P_{10}$.

$$ \text{Denominator} = 63.831 - 58.983 $$ $$ \text{Denominator} = 4.848 $$

Step 3: Calculate Kelly's Coefficient

Now, divide the numerator by the denominator.

$$ \text{Sk} = \frac{\text{Numerator}}{\text{Denominator}} $$ $$ \text{Sk} = \frac{0.124}{4.848} $$ $$ \text{Sk} \approx 0.0255735561 $$

Rounding the result to three decimal places, we get:

$$ \text{Sk} \approx 0.026 $$

Interpreting the Calculated Kelly's Coefficient

The calculated value for Kelly's coefficient of skewness is approximately 0.026. This value is positive and close to zero, suggesting that the distribution is slightly positively skewed. The right tail of the distribution is slightly longer or fatter than the left tail.

Additional Information: Percentiles in Data Analysis

Percentiles are values that divide a dataset into 100 equal parts. The nth percentile is the value below which n percent of the observations fall. Key percentiles include:

• $P_{10}$: The value below which 10% of the data lies.
• $P_{50}$: The value below which 50% of the data lies. This is the median.
• $P_{90}$: The value below which 90% of the data lies.

The range between $P_{10}$ and $P_{90}$ covers the central 80% of the data. Kelly's coefficient uses these specific percentiles to avoid being heavily influenced by the extreme ends of the distribution, which are more susceptible to outliers. This makes it a robust measure of skewness.