The first four raw moments of distribution are 2, 136, 320, and 40,000, The coefficient of skewness is:
Calculating Coefficient of Skewness from Raw Moments
To find the coefficient of skewness from raw moments, we first need to convert the raw moments into central moments. Central moments are moments about the mean, and they are directly used in formulas for measures like skewness and kurtosis.
Understanding Raw and Central Moments
Raw moments (denoted by \( \mu'_k \)) are moments taken about the origin (zero). The first raw moment \( \mu'_1 \) is the mean of the distribution.
Central moments (denoted by \( \mu_k \)) are moments taken about the mean. The first central moment \( \mu_1 \) is always zero. The second central moment \( \mu_2 \) is the variance. The third central moment \( \mu_3 \) is related to skewness, and the fourth central moment \( \mu_4 \) is related to kurtosis.
Given Raw Moments
The first four raw moments of the distribution are given as:
- \( \mu'_1 = 2 \)
- \( \mu'_2 = 136 \)
- \( \mu'_3 = 320 \)
- \( \mu'_4 = 40000 \)
Converting Raw Moments to Central Moments
We use the following formulas to convert raw moments to central moments:
- \( \mu_1 = 0 \) (by definition)
- \( \mu_2 = \mu'_2 - (\mu'_1)^2 \)
- \( \mu_3 = \mu'_3 - 3\mu'_2\mu'_1 + 2(\mu'_1)^3 \)
- \( \mu_4 = \mu'_4 - 4\mu'_3\mu'_1 + 6\mu'_2(\mu'_1)^2 - 3(\mu'_1)^4 \)
Let's calculate the central moments:
- Second Central Moment (\( \mu_2 \)): \[ \mu_2 = \mu'_2 - (\mu'_1)^2 = 136 - (2)^2 = 136 - 4 = 132 \]
- Third Central Moment (\( \mu_3 \)): \[ \mu_3 = \mu'_3 - 3\mu'_2\mu'_1 + 2(\mu'_1)^3 = 320 - 3(136)(2) + 2(2)^3 = 320 - 816 + 2(8) = 320 - 816 + 16 = -480 \]
- Fourth Central Moment (\( \mu_4 \)): \[ \mu_4 = \mu'_4 - 4\mu'_3\mu'_1 + 6\mu'_2(\mu'_1)^2 - 3(\mu'_1)^4 \] \[ \mu_4 = 40000 - 4(320)(2) + 6(136)(2)^2 - 3(2)^4 \] \[ \mu_4 = 40000 - 2560 + 6(136)(4) - 3(16) \] \[ \mu_4 = 40000 - 2560 + 3264 - 48 \] \[ \mu_4 = 37440 + 3216 = 40656 \]
So, the calculated central moments are \( \mu_2 = 132 \), \( \mu_3 = -480 \), and \( \mu_4 = 40656 \).
Calculating the Coefficient of Skewness
The coefficient of skewness based on moments is often represented by \( \beta_1 \) or \( \gamma_1 \). Karl Pearson's Beta coefficient of skewness is given by:
\[ \beta_1 = \dfrac{\mu_3^2}{\mu_2^3} \]This measure indicates the degree of asymmetry of the distribution. A value of 0 indicates symmetry, a positive value indicates positive skew (tail to the right), and a negative value is not possible for \( \beta_1 \) because \( \mu_3 \) is squared.
Karl Pearson's Gamma coefficient of skewness is given by \( \gamma_1 = \sqrt{\beta_1} = \dfrac{\mu_3}{(\sqrt{\mu_2})^3} = \dfrac{\mu_3}{\mu_2^{3/2}} \). This measure retains the sign of \( \mu_3 \) and thus indicates the direction of skewness.
Looking at the options provided, they are in the form of \( \mu_3^2 / \mu_2^3 \) or similar structure, suggesting we should calculate \( \beta_1 \).
Using our calculated central moments \( \mu_2 = 132 \) and \( \mu_3 = -480 \):
\[ \beta_1 = \dfrac{(-480)^2}{(132)^3} \]Comparing with Options
Let's compare our result with the given options:
| Option | Expression | Matches Calculation? |
|---|---|---|
| 1 | \( \dfrac{40656}{(132)^2} \) | No (Uses \( \mu_4 \) and \( \mu_2^2 \)) |
| 2 | \( \dfrac{(40656)^2}{(132)^3} \) | No (Uses \( \mu_4^2 \) and \( \mu_2^3 \)) |
| 3 | \( \dfrac{-480}{(132)^2} \) | No (Uses \( \mu_3 \) and \( \mu_2^2 \)) |
| 4 | \( \dfrac{(-480)^2}{(132)^3} \) | Yes (Matches \( \mu_3^2 / \mu_2^3 \)) |
Our calculated value for the coefficient of skewness \( \beta_1 \) is \( \dfrac{(-480)^2}{(132)^3} \), which matches Option 4.
Revision Table: Moments and Skewness
| Concept | Notation | Definition/Formula (Central Moments from Raw) |
|---|---|---|
| First Raw Moment (Mean) | \( \mu'_1 \) | Given |
| Second Raw Moment | \( \mu'_2 \) | Given |
| Third Raw Moment | \( \mu'_3 \) | Given |
| Fourth Raw Moment | \( \mu'_4 \) | Given |
| First Central Moment | \( \mu_1 \) | \( 0 \) |
| Second Central Moment (Variance) | \( \mu_2 \) | \( \mu'_2 - (\mu'_1)^2 \) |
| Third Central Moment | \( \mu_3 \) | \( \mu'_3 - 3\mu'_2\mu'_1 + 2(\mu'_1)^3 \) |
| Fourth Central Moment | \( \mu_4 \) | \( \mu'_4 - 4\mu'_3\mu'_1 + 6\mu'_2(\mu'_1)^2 - 3(\mu'_1)^4 \) |
| Beta Coefficient of Skewness | \( \beta_1 \) | \( \dfrac{\mu_3^2}{\mu_2^3} \) |
| Gamma Coefficient of Skewness | \( \gamma_1 \) | \( \dfrac{\mu_3}{\sqrt{\mu_2^3}} \) |
Additional Information on Skewness and Moments
Skewness is a measure of the asymmetry in a probability distribution. A distribution is symmetric if it looks the same on both sides of the mean. If the distribution is skewed, the data is spread out more on one side of the mean than the other.
- Positive Skewness: The tail on the right side is longer or fatter than the tail on the left side. The mean is typically greater than the median.
- Negative Skewness: The tail on the left side is longer or fatter than the tail on the right side. The mean is typically less than the median.
- Zero Skewness: The distribution is symmetric. The mean, median, and mode are often equal (though not always for all symmetric distributions).
Moments are fundamental descriptive statistics used to characterize the shape and properties of a distribution. Raw moments and central moments are commonly used, but moments about any arbitrary point can also be calculated.
The third central moment \( \mu_3 \) directly measures the asymmetry. If \( \mu_3 > 0 \), the distribution is positively skewed. If \( \mu_3 < 0 \), it's negatively skewed. If \( \mu_3 = 0 \), it's symmetric.
The coefficient \( \beta_1 = \mu_3^2 / \mu_2^3 \) is a pure number, independent of the units of the variable, making it useful for comparing the degree of skewness across different distributions. Its square root, \( \gamma_1 \), is also unit-free and its sign matches the sign of \( \mu_3 \).
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