If the first, fifth and ninth decile of frequency distribution x i|f iare 3, 10, 16, respectively, then Kelly's coefficient of skewness is:
Understanding Kelly's Coefficient of Skewness
Kelly's coefficient of skewness is a measure used to determine the asymmetry of a probability distribution. It is based on percentiles or deciles, which are positional measures. Unlike the Pearson coefficient which uses mean and standard deviation, Kelly's coefficient uses quantiles, making it less sensitive to extreme values.
The formula for Kelly's coefficient of skewness ($S_k$) using deciles is given by:
\(S_k = \frac{D_1 + D_9 - 2D_5}{D_9 - D_1}\)
Where:
- \(D_1\) is the first decile (10th percentile)
- \(D_5\) is the fifth decile (50th percentile or median)
- \(D_9\) is the ninth decile (90th percentile)
Applying the Formula to the Given Deciles
We are provided with the following values from the frequency distribution:
- First decile, \(D_1 = 3\)
- Fifth decile, \(D_5 = 10\)
- Ninth decile, \(D_9 = 16\)
Now, we will substitute these values into Kelly's coefficient of skewness formula:
\(S_k = \frac{D_1 + D_9 - 2D_5}{D_9 - D_1}\)
\(S_k = \frac{3 + 16 - 2 \times 10}{16 - 3}\)
\(S_k = \frac{19 - 20}{13}\)
\(S_k = \frac{-1}{13}\)
Calculating Kelly's Skewness Result
The calculation results in a value of \(-\frac{1}{13}\). This negative value indicates that the distribution is negatively skewed (or left-skewed). In a negatively skewed distribution, the tail on the left side of the distribution is longer or fatter than the tail on the right side.
The calculated Kelly's coefficient of skewness for the given frequency distribution is \(-\frac{1}{13}\).
Revision Table: Key Statistical Concepts
| Concept | Description |
|---|---|
| Deciles | Divide a dataset into ten equal parts. \(D_1\) is the 10th percentile, \(D_5\) is the 50th percentile (Median), \(D_9\) is the 90th percentile. |
| Skewness | A measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. |
| Kelly's Coefficient of Skewness | A non-parametric measure of skewness based on deciles (or other quantiles like percentiles or quartiles). |
| Negative Skewness | The tail on the left side of the distribution is longer; mean < median < mode (typically). |
| Positive Skewness | The tail on the right side of the distribution is longer; mean > median > mode (typically). |
Additional Information on Skewness and Deciles
Skewness is a fundamental concept in descriptive statistics that helps us understand the shape of a data distribution. A symmetrical distribution has a skewness of zero. Distributions that are not symmetrical are skewed.
Deciles are part of a broader family of measures called quantiles. Quantiles divide a dataset into equal parts. Other common quantiles include quartiles (dividing into four parts) and percentiles (dividing into one hundred parts).
Kelly's coefficient of skewness offers an alternative to measures like Pearson's coefficient, which can be heavily influenced by outliers. By using deciles ($D_1, D_5, D_9$) which are less sensitive to extreme values compared to the mean or standard deviation, Kelly's coefficient provides a robust measure of skewness.
The formula for Kelly's coefficient can also be expressed using percentiles:
\(S_k = \frac{P_{10} + P_{90} - 2P_{50}}{P_{90} - P_{10}}\)
Where \(P_{10}\), \(P_{50}\), and \(P_{90}\) are the 10th, 50th, and 90th percentiles, respectively. Since the 10th percentile is the first decile ($D_1$), the 50th percentile is the fifth decile ($D_5$), and the 90th percentile is the ninth decile ($D_9$), the formulas are equivalent.
Understanding the value of Kelly's coefficient:
- If \(S_k > 0\), the distribution is positively skewed.
- If \(S_k < 0\), the distribution is negatively skewed.
- If \(S_k = 0\), the distribution is symmetrical.
In this problem, we found \(S_k = -\frac{1}{13}\), which is less than 0, confirming the distribution is negatively skewed.
Similar Questions
Positive skewness means that the frequencies in the distribution are spread:
Which of the following is an absolute measure of skewness?
For a negatively skewed and platykurtic distribution:
If the distribution is negatively skewed then:
If median = 45, mean = 39.27, SD = 22.81, then the coefficient of Skewness is:
If Mean > Median > Mode, then distribution is:
For a data set, the following information is obtained. If Q1 = 62, Q2 = 142 and Q3 = 195, then Bowley’s coefficient of skewness is:
In case of a moderately skewed frequency distribution, the mean is 50 and the median is 53. If the coefficient of variation is 20%, then the coefficient of skewness is:
For a symmetrical distribution, we have:
The mode of the given data set is 12. The sum of the frequencies on both sides of mode are 16. The skewness:
Important Questions from Measures of skewness
Positive skewness means that the frequencies in the distribution are spread:
Which of the following is an absolute measure of skewness?
For a negatively skewed and platykurtic distribution:
If the distribution is negatively skewed then:
If median = 45, mean = 39.27, SD = 22.81, then the coefficient of Skewness is:
