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:

1. Using Mean and Mode: \(Sk = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}\)
2. 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

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

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.