Calculating Coefficients of Skewness and Kurtosis Given Central Moments

In statistics, the coefficients of skewness and kurtosis are widely used to understand the distribution of a variable. These coefficients provide important insights into the symmetry and the shape of the distribution, respectively. This article will guide you through a detailed process to calculate these coefficients when given the central moments of a variable.

Background on Moments

Central moments are statistical measures that describe various aspects of the distribution of a variable. The central moments about a point (in this case, value 3) provide crucial information for computing skewness and kurtosis.

Given Data

Moments about the Value 3:

First Central Moment (μ?): 1.7 Second Central Moment (μ?): 8.9 Third Central Moment (μ?): 39.5 Fourth Central Moment (μ?): 211.7

Calculating Central Moments

To derive these moments, we start with the given moments and use the formulas for central moments. Central moments are derived from the raw moments by subtracting the appropriate powers of the mean.

First Central Moment (μ?)

The first central moment is zero by definition because it represents the deviation from the mean. Since it is given directly, we use:

[ μ? 1.7 ]

Second Central Moment (μ?)

The second central moment, which is the variance, is calculated as:

[ μ? 8.9 ]

Third Central Moment (μ?)

For the third central moment, we apply the formula:

[ μ? 39.5 ]

Fourth Central Moment (μ?)

The fourth central moment is provided directly:

[ μ? 211.7 ]

Computing Skewness and Kurtosis

Now that we have the central moments, we can proceed to calculate the coefficients of skewness and kurtosis.

Skewness Formula

Skewness (γ?) can be computed using the third central moment (μ?) and the second central moment (μ?) raised to the power of 3/2:

[ γ? frac{μ?}{μ?^{3/2}} ]

Substituting the given values:

[ γ? frac{39.5}{8.9^{3/2}} ]

Calculate the denominator first:

[ 8.9^{3/2} (8.9)^{1.5} approx 25.27 ]

Now, compute the skewness:

[ γ? frac{39.5}{25.27} approx 1.56 ]

Kurtosis Formula

Kurtosis (γ?) is computed using the fourth central moment (μ?) and the second central moment (μ? squared):

[ γ? frac{μ? - 3μ?μ? 3μ?μ?2 - μ?3}{μ?2} ]

Substituting the given values:

[ γ? frac{211.7 - 3(8.9)(39.5) 3(1.7)(8.9)2 - (1.7)3}{8.92} ]

Calculate each term separately:

[ 3(8.9)(39.5) 1143.15 ] [ 3(1.7)(8.9)2 3(1.7)(79.21) 405.14 ] [ (1.7)3 4.91 ] [ 211.7 - 1143.15 405.14 - 4.91 -541.22 ] [ (8.9)2 79.21 ] [ γ? frac{-541.22}{79.21} approx -6.81 ]

Thus, the skewness and kurtosis values for the given data are:

Skewness: γ? ≈ 1.56 Kurtosis: γ? ≈ -6.81

Explanation of Results

A positive skewness (γ?) indicates a right-skewed distribution, while negative skewness suggests a left-skewed distribution. In this example, the skewness value of approximately 1.56 indicates a right-skewed distribution.

A negative kurtosis value (γ?) suggests a flatter distribution compared to a normal distribution. The calculated value of -6.81 indicates a heavy-tailed distribution, meaning the data points are more spread out from the mean than a normal distribution would be.

Conclusion

This article has provided a step-by-step guide on how to calculate the coefficients of skewness and kurtosis using the given central moments. Understanding these measures is crucial for analyzing the distribution of variables and making informed decisions based on their statistical properties.