Let's analyze the given information about the data set and its mode.

We are given:

• The mode of the data set is 12.
• The sum of the frequencies on both sides of the mode is 16.

We are asked to determine the skewness of this data set based on this information.

Understanding Mode and Skewness in Data Analysis

The mode is the value that appears most frequently in a data set. In a frequency distribution, it is the observation with the highest frequency.

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It indicates the direction and magnitude of a distribution's departure from symmetry.

• A symmetrical distribution has zero skewness (e.g., normal distribution). In a symmetrical distribution, the mean, median, and mode are often equal.
• A positively skewed distribution (right-skewed) has a tail extending towards the right. The mean is typically greater than the median, which is greater than the mode.
• A negatively skewed distribution (left-skewed) has a tail extending towards the left. The mean is typically less than the median, which is less than the mode.

Methods for Calculating Skewness

There are several ways to calculate skewness. Some common methods include:

1. Pearson's First Coefficient of Skewness: This is based on the relationship between the mean, mode, and standard deviation.
   \[Skewness = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}\]
2. Pearson's Second Coefficient of Skewness: This is used when the mode is not well-defined, and it involves the median.
   \[Skewness = \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}}\]
3. Bowley's Coefficient of Skewness: This is based on quartiles.
   \[Skewness = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}\]
   where \(Q_1\), \(Q_2\), and \(Q_3\) are the first, second (median), and third quartiles, respectively.

Analyzing the Given Information

We are given the mode (12) and the sum of frequencies on both sides of the mode (16). This information tells us that the value 12 is the mode, meaning it has the highest frequency. The sum of frequencies for all data values other than 12 is 16.

To calculate skewness using any of the standard formulas mentioned above, we need more information about the data set. Specifically, we need to know:

• The actual data values (or the full frequency distribution).
• The frequency of the mode (the value 12).
• The frequencies of the other data values.

From the given information (mode = 12, sum of frequencies on both sides of mode = 16), we cannot determine the following crucial statistics needed for skewness calculation:

• The Mean of the data set.
• The Median of the data set.
• The Standard Deviation of the data set.
• The Quartiles ($Q_1$, $Q_2$, $Q_3$) of the data set.

Knowing the mode and the sum of frequencies on either side is not enough to reconstruct the entire frequency distribution or calculate these required measures. For example, a data set with frequencies [5, 8, 10 (at 12), 6, 5] has a mode of 12 and sum of frequencies on both sides = 5+8+6+5 = 24. The current problem gives sum = 16, but we don't know how that 16 is distributed among other values, nor do we know the frequency of the mode itself, nor the other values in the data set.

Since we cannot calculate any measure of skewness using the provided data, the skewness cannot be determined from this information. Therefore, the skewness does not exist (or cannot be calculated) based on the given data.

Conclusion

The given information (mode = 12 and sum of frequencies on both sides of mode = 16) is insufficient to calculate the skewness of the data set. We need more information about the data distribution (like mean, median, standard deviation, or the full frequency distribution) to compute skewness. Hence, based on the provided data points, the skewness cannot be determined.

Since essential components for calculating skewness are missing, we conclude that the skewness does not exist based on the limited information provided.

Revision Table: Key Concepts for Data Analysis

Additional Information: Why More Data is Needed for Skewness

Skewness describes the shape of the distribution. Imagine trying to describe the shape of a mountain range just by knowing the height of its highest peak (the mode) and the total number of trees growing on the slopes below that peak (sum of frequencies on both sides). You might know where the highest point is, but you wouldn't know how steep the slopes are, where the valleys are, or how the elevation changes away from the peak. Similarly, knowing only the mode and the sum of other frequencies doesn't tell you if the data points are spread out symmetrically around the mode or if they trail off more on one side than the other. Without the full distribution, calculating measures like the mean, median, or standard deviation—which capture the location of the center and the spread of the data—is impossible. These measures are fundamental to calculating skewness.