If a variable has three values -k, 0 and 3k with corresponding frequencies as 3k, 2k and k respectively, then the value of coefficient of skewness b1 is :
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Calculating Coefficient of Skewness \(b_1\) for a Frequency Distribution
To find the value of the coefficient of skewness \(b_1\), we need to calculate the mean, variance, and the third central moment of the given frequency distribution. The coefficient of skewness \(b_1\) is related to the third central moment (\(\mu_3\)) and the standard deviation (\(\sigma\)).
Understanding Coefficient of Skewness \(b_1\)
\(b_1\) is a measure of skewness derived from the third moment. It is defined as the square of the Pearson moment coefficient of skewness (\(\gamma_1\)):
\( b_1 = \left(\frac{\mu_3}{\sigma^3}\right)^2 \)
Where:
- \(\mu_3\) is the third central moment, calculated as \( \mu_3 = \frac{\sum f(x - \bar{x})^3}{\sum f} \).
- \(\sigma\) is the standard deviation, calculated as \( \sigma = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}} \).
- \(\bar{x}\) is the mean, calculated as \( \bar{x} = \frac{\sum fx}{\sum f} \).
Given Data
We are given the following variable values and their corresponding frequencies:
| Variable (x) | Frequency (f) |
|---|---|
| -k | 3k |
| 0 | 2k |
| 3k | k |
Step-by-Step Calculation of \(b_1\) Coefficient
Let's calculate the necessary components:
- Calculate Total Frequency (\(\sum f\)):
\( \sum f = 3k + 2k + k = 6k \)
- Calculate Sum of \(fx\) (\(\sum fx\)):
\( \sum fx = (-k)(3k) + (0)(2k) + (3k)(k) \)
\( \sum fx = -3k^2 + 0 + 3k^2 = 0 \)
- Calculate Mean (\(\bar{x}\)):
\( \bar{x} = \frac{\sum fx}{\sum f} = \frac{0}{6k} = 0 \)
The mean of the distribution is 0.
- Calculate Sum of \(fx^2\) (\(\sum fx^2\)):
\( \sum fx^2 = (-k)^2(3k) + (0)^2(2k) + (3k)^2(k) \)
\( \sum fx^2 = (k^2)(3k) + (0)(2k) + (9k^2)(k) \)
\( \sum fx^2 = 3k^3 + 0 + 9k^3 = 12k^3 \)
- Calculate Variance (\(\sigma^2\)):
Since the mean (\(\bar{x}\)) is 0, the variance is \(\sigma^2 = \frac{\sum f(x - \bar{x})^2}{\sum f} = \frac{\sum fx^2}{\sum f}\).
\( \sigma^2 = \frac{12k^3}{6k} = 2k^2 \)
- Calculate Standard Deviation (\(\sigma\)):
\( \sigma = \sqrt{\sigma^2} = \sqrt{2k^2} = k\sqrt{2} \) (Assuming \(k > 0\), as it's a frequency multiplier)
- Calculate Sum of \(fx^3\) (\(\sum fx^3\)):
\( \sum fx^3 = (-k)^3(3k) + (0)^3(2k) + (3k)^3(k) \)
\( \sum fx^3 = (-k^3)(3k) + (0)(2k) + (27k^3)(k) \)
\( \sum fx^3 = -3k^4 + 0 + 27k^4 = 24k^4 \)
- Calculate Third Central Moment (\(\mu_3\)):
Since the mean (\(\bar{x}\)) is 0, \(\mu_3 = \frac{\sum f(x - \bar{x})^3}{\sum f} = \frac{\sum fx^3}{\sum f}\).
\( \mu_3 = \frac{24k^4}{6k} = 4k^3 \)
- Calculate Pearson Moment Coefficient of Skewness (\(\gamma_1\)):
\( \gamma_1 = \frac{\mu_3}{\sigma^3} \)
First, calculate \(\sigma^3\):
\( \sigma^3 = (k\sqrt{2})^3 = k^3 (\sqrt{2})^3 = k^3 (2\sqrt{2}) = 2\sqrt{2}k^3 \)
Now, calculate \(\gamma_1\):
\( \gamma_1 = \frac{4k^3}{2\sqrt{2}k^3} = \frac{4}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} \)
- Calculate Coefficient of Skewness \(b_1\):
\( b_1 = \gamma_1^2 = (\sqrt{2})^2 = 2 \)
Result
The value of the coefficient of skewness \(b_1\) is 2.
Revision Table: Statistical Measures Summary
| Measure | Formula (for grouped data, \(\bar{x}=0\)) | Calculated Value |
|---|---|---|
| Total Frequency | \(\sum f\) | \(6k\) |
| Mean (\(\bar{x}\)) | \(\frac{\sum fx}{\sum f}\) | \(0\) |
| Variance (\(\sigma^2\)) | \(\frac{\sum fx^2}{\sum f} - \bar{x}^2\) | \(2k^2\) |
| Standard Deviation (\(\sigma\)) | \(\sqrt{\sigma^2}\) | \(k\sqrt{2}\) |
| Third Central Moment (\(\mu_3\)) | \(\frac{\sum f(x - \bar{x})^3}{\sum f} = \frac{\sum fx^3}{\sum f}\) | \(4k^3\) |
| Coefficient of Skewness \(b_1\) | \(\left(\frac{\mu_3}{\sigma^3}\right)^2\) | \(2\) |
Additional Information: Skewness in Statistics
Skewness is a key concept in statistics that quantifies the degree and direction of asymmetry of a probability distribution.
- A symmetric distribution has zero skewness (e.g., Normal distribution).
- A distribution with a long tail towards the right is positively skewed (\(\gamma_1 > 0\)). The bulk of the data is on the left.
- A distribution with a long tail towards the left is negatively skewed (\(\gamma_1 < 0\)). The bulk of the data is on the right.
The moment coefficient of skewness (\(\gamma_1\)) uses the third moment about the mean. It is sensitive to outliers and extreme values, especially in the tails of the distribution.
The coefficient \(b_1 = \gamma_1^2\) is another measure of skewness. Since it's a square, it only tells us about the magnitude of skewness, not its direction. A value of \(b_1 = 0\) indicates a symmetric distribution (\(\gamma_1=0\)). Higher values of \(b_1\) suggest greater asymmetry.
Other measures of skewness exist, such as Pearson's coefficients (based on mean-mode or mean-median) and Bowley's coefficient (based on quartiles). Each measure has its own properties and uses depending on the data and the specific analysis goal.
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