Understanding Population Skewness for Grouped Data

The question asks us to determine the population skewness for a given set of grouped data. Skewness is a statistical measure that describes the asymmetry of the probability distribution of a real-valued random variable about its mean. A distribution is symmetrical if it looks the same on both sides of the mean. Positive skewness indicates a tail extending towards higher values, while negative skewness indicates a tail extending towards lower values. Zero skewness suggests a symmetrical distribution.

Given Grouped Data

We are provided with the following grouped data:

Calculating Population Skewness

To calculate the population skewness for grouped data, one common method is to use the third central moment. The formula for the population skewness coefficient (\(\gamma_1\)) based on moments is:

\(\gamma_1 = \frac{\mu_3}{\sigma^3}\)

where \(\mu_3\) is the third central moment and \(\sigma\) is the population standard deviation.

The third central moment (\(\mu_3\)) for grouped data is approximated by:

\(\mu_3 = \frac{\sum f_i (m_i - \bar{x})^3}{N}\)

where:

• \(f_i\) is the frequency of the \(i\)-th class.
• \(m_i\) is the midpoint of the \(i\)-th class.
• \(\bar{x}\) is the population mean.
• \(N\) is the total number of observations (\(\sum f_i\)).

First, let's find the midpoints (\(m_i\)) for each class:

• Class 1-3: \(m_1 = \frac{1+3}{2} = 2\)
• Class 3-5: \(m_2 = \frac{3+5}{2} = 4\)
• Class 5-7: \(m_3 = \frac{5+7}{2} = 6\)

Next, we calculate the total frequency \(N\):

\(N = 2 + 1 + 2 = 5\)

Now, let's calculate the population mean (\(\bar{x}\)):

\(\bar{x} = \frac{\sum f_i m_i}{N}\)

• \(f_1 m_1 = 2 \times 2 = 4\)
• \(f_2 m_2 = 1 \times 4 = 4\)
• \(f_3 m_3 = 2 \times 6 = 12\)

\(\sum f_i m_i = 4 + 4 + 12 = 20\)

\(\bar{x} = \frac{20}{5} = 4\)

The population mean is 4.

Now, we calculate the terms \((m_i - \bar{x})^3\) and \(f_i (m_i - \bar{x})^3\):

• For class 1-3: \(m_1 - \bar{x} = 2 - 4 = -2\). \((m_1 - \bar{x})^3 = (-2)^3 = -8\). \(f_1 (m_1 - \bar{x})^3 = 2 \times (-8) = -16\).
• For class 3-5: \(m_2 - \bar{x} = 4 - 4 = 0\). \((m_2 - \bar{x})^3 = (0)^3 = 0\). \(f_2 (m_2 - \bar{x})^3 = 1 \times 0 = 0\).
• For class 5-7: \(m_3 - \bar{x} = 6 - 4 = 2\). \((m_3 - \bar{x})^3 = (2)^3 = 8\). \(f_3 (m_3 - \bar{x})^3 = 2 \times 8 = 16\).

Calculate the sum \(\sum f_i (m_i - \bar{x})^3\):

\(\sum f_i (m_i - \bar{x})^3 = -16 + 0 + 16 = 0\)

Now, calculate the third central moment (\(\mu_3\)):

\(\mu_3 = \frac{\sum f_i (m_i - \bar{x})^3}{N} = \frac{0}{5} = 0\)

Since the third central moment (\(\mu_3\)) is 0, the population skewness (\(\gamma_1\)) is also 0, provided the standard deviation (\(\sigma\)) is not zero.

Let's quickly calculate the population variance (\(\sigma^2\)) to confirm \(\sigma \neq 0\):

\(\sigma^2 = \frac{\sum f_i (m_i - \bar{x})^2}{N}\)

• For class 1-3: \((m_1 - \bar{x})^2 = (-2)^2 = 4\). \(f_1 (m_1 - \bar{x})^2 = 2 \times 4 = 8\).
• For class 3-5: \((m_2 - \bar{x})^2 = (0)^2 = 0\). \(f_2 (m_2 - \bar{x})^2 = 1 \times 0 = 0\).
• For class 5-7: \((m_3 - \bar{x})^2 = (2)^2 = 4\). \(f_3 (m_3 - \bar{x})^2 = 2 \times 4 = 8\).

\(\sum f_i (m_i - \bar{x})^2 = 8 + 0 + 8 = 16\)

\(\sigma^2 = \frac{16}{5} = 3.2\)

\(\sigma = \sqrt{3.2}\)

Since \(\sigma^2 = 3.2 \neq 0\), \(\sigma \neq 0\). Therefore, the population skewness is indeed 0.

A skewness of zero indicates that the distribution is symmetrical around its mean.

Analyzing the Options

Based on our calculation, the population skewness is 0.

• Option 1: is zero
• Option 2: is negative
• Option 3: is positive
• Option 4: cannot be computed

Our calculated value matches Option 1.

Conclusion on Population Skewness

The calculated population skewness for the given grouped data is zero. This means the distribution of the data is symmetrical.

Revision Table: Key Skewness Concepts

Additional Information: Measures of Skewness

Besides the moment-based coefficient (\(\gamma_1\)), other measures of skewness exist:

• Pearson's First Coefficient of Skewness: \(Sk_1 = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}\). Useful when a clear mode exists.
• Pearson's Second Coefficient of Skewness: \(Sk_2 = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}}\). Often used when the mode is not well-defined.
• Bowley's Skewness (Quantile Skewness): Based on quartiles, \(Sk_B = \frac{(Q_3 - Q_2) - (Q_2 - Q_1)}{Q_3 - Q_1} = \frac{Q_1 + Q_3 - 2Q_2}{Q_3 - Q_1}\). This measure is robust to extreme values.

All these measures aim to quantify the degree and direction of asymmetry in a distribution. The choice of measure depends on the data and the specific context.