If the mean, mode and quartile deviation of a distribution is 2, 7 and 3, respectively, then Karl Pearson's coefficient of skewness is given by:
-10/9
Understanding Karl Pearson's Coefficient of Skewness
Karl Pearson's coefficient of skewness is a common measure used to understand the asymmetry of a probability distribution. It indicates whether the data is skewed to the left (negatively skewed) or to the right (positively skewed), or if it is symmetrical.
There are two main formulas for Karl Pearson's coefficient of skewness:
- Using Mean and Mode: \(Sk = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}\)
- Using Mean and Median: \(Sk = \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}}\)
In this question, we are given the mean, mode, and quartile deviation. This suggests we should use the first formula involving the mean and mode. However, the formula requires the standard deviation (SD), which is not directly provided. We are given the quartile deviation (QD).
Using Quartile Deviation to Estimate Standard Deviation
For many distributions, especially those that are roughly symmetrical or unimodal, there's an approximate relationship between the Quartile Deviation and the Standard Deviation. A common approximation, particularly related to the normal distribution, is:
\(QD \approx \frac{2}{3} \times SD\)
From this, we can estimate the Standard Deviation:
\(SD \approx \frac{3}{2} \times QD\)
Given the quartile deviation (QD) is 3, we can estimate the standard deviation:
\(SD \approx \frac{3}{2} \times 3 = \frac{9}{2} = 4.5\)
Calculating Karl Pearson's Coefficient of Skewness
Now we have the necessary components to calculate Karl Pearson's coefficient of skewness using the formula \(Sk = \frac{\text{Mean} - \text{Mode}}{SD}\).
Given values:
- Mean (\(\bar{x}\)) = 2
- Mode (Mo) = 7
- Quartile Deviation (QD) = 3
Estimated Standard Deviation (SD) = 4.5
Let's plug the values into the formula:
\(Sk = \frac{\text{Mean} - \text{Mode}}{SD}\)
\(Sk = \frac{2 - 7}{4.5}\)
\(Sk = \frac{-5}{4.5}\)
To simplify the fraction, we can write 4.5 as \(\frac{9}{2}\):
\(Sk = \frac{-5}{9/2}\)
\(Sk = -5 \times \frac{2}{9}\)
\(Sk = \frac{-10}{9}\)
The calculated Karl Pearson's coefficient of skewness is \(-10/9\).
Summary of Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify Given Values | Mean=2, Mode=7, QD=3 |
| 2 | Estimate SD using QD | \(SD \approx \frac{3}{2} \times QD = \frac{3}{2} \times 3 = 4.5\) |
| 3 | Calculate Skewness | \(Sk = \frac{\text{Mean} - \text{Mode}}{SD} = \frac{2 - 7}{4.5}\) |
| 4 | Simplify Result | \(Sk = \frac{-5}{4.5} = \frac{-5}{9/2} = -5 \times \frac{2}{9} = \frac{-10}{9}\) |
This result, \(-10/9\), matches one of the provided options. The negative value indicates that the distribution is negatively skewed, meaning it has a longer tail on the left side.
Revision Table: Key Statistical Measures
| Measure | Description | How it Describes Distribution |
|---|---|---|
| Mean | Average value | Center of mass |
| Median | Middle value | Center of position (50th percentile) |
| Mode | Most frequent value | Peak of the distribution |
| Standard Deviation | Spread of data points around the mean | Variability or dispersion |
| Quartile Deviation | Half of the Interquartile Range (IQR) | Spread of the middle 50% of data |
| Skewness | Asymmetry of the distribution | Shape (left-skewed, right-skewed, symmetrical) |
Additional Information: Skewness Interpretation
Karl Pearson's coefficient of skewness helps us understand the shape of the distribution in terms of its symmetry. Here's a simple interpretation:
- If \(Sk > 0\): The distribution is positively skewed (skewed to the right). The tail is longer on the right side. Mean > Median > Mode (typically).
- If \(Sk < 0\): The distribution is negatively skewed (skewed to the left). The tail is longer on the left side. Mean < Median < Mode (typically).
- If \(Sk = 0\): The distribution is symmetrical. Mean = Median = Mode (for unimodal distributions).
It's important to remember that the relationship \(SD \approx \frac{3}{2} \times QD\) is an approximation often used in contexts like the normal distribution. Using this approximation is necessary here because the standard deviation is not directly given, but the quartile deviation is, alongside the mean and mode required for Karl Pearson's first formula.
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