Analyzing Skewness for Grouped Data

Let's analyze the provided grouped data to determine its population skewness. Skewness is a measure of the asymmetry of a probability distribution. It tells us if the data is skewed to the left (negative skew), skewed to the right (positive skew), or symmetric (zero skew).

Understanding the Grouped Data

The data is given in classes with corresponding frequencies:

To analyze the distribution of this grouped data, we typically use the midpoint of each class to represent the values within that class.

Calculating Class Midpoints

The midpoint of a class interval \([L, U]\) is calculated as \(\frac{L + U}{2}\), where \(L\) is the lower limit and \(U\) is the upper limit.

• For the class 2-4: Midpoint = \(\frac{2 + 4}{2} = \frac{6}{2} = 3\)
• For the class 4-6: Midpoint = \(\frac{4 + 6}{2} = \frac{10}{2} = 5\)
• For the class 6-8: Midpoint = \(\frac{6 + 8}{2} = \frac{14}{2} = 7\)

Examining the Distribution Shape

Now let's look at the midpoints and their corresponding frequencies:

We can visualize this distribution:

• At midpoint 3, there are 2 observations.
• At midpoint 5, there is 1 observation.
• At midpoint 7, there are 2 observations.

The distribution has its highest frequencies (2) at the extreme midpoints (3 and 7) and a lower frequency (1) at the center midpoint (5). If we plot the frequency against the midpoint, we would see a shape that is symmetric around the midpoint 5. The distribution is balanced, with the same frequency (2) on the left side (midpoint 3) as on the right side (midpoint 7) relative to the central frequency (1 at midpoint 5).

Determining Population Skewness

A distribution is symmetric if the left side is a mirror image of the right side. For symmetric distributions, the mean, median, and mode are often equal (though not always required for symmetry itself). A key characteristic of symmetric distributions is that they have zero skewness.

In this case, the frequency distribution is perfectly symmetric around the central value (represented by midpoint 5). Therefore, the population skewness for this grouped data is zero.

Revision Table: Key Concepts

Additional Information on Skewness Calculation

While we determined skewness by observing the symmetry here, skewness can also be calculated mathematically. A common measure is Pearson's coefficient of skewness or the moment coefficient of skewness.

Pearson's First Coefficient of Skewness (for unimodal distribution):

\(SK_1 = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}\)

Pearson's Second Coefficient of Skewness (generally applicable):

\(SK_2 = \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}}\)

The moment coefficient of skewness (\(\gamma_1\)) is based on the third standardized moment:

\(\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right]\)

For grouped data, these measures require calculating the mean, median, mode, and standard deviation from the frequency distribution, using the midpoints to represent the class values. However, recognizing symmetry provides a direct conclusion about zero skewness without needing these calculations.