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The mean, also known as the average, is a measure of central tendency that provides an indication of the overall level of a dataset. Calculating the mean is a straightforward process. You sum up all the values in your dataset and then divide by the number of values. This calculation gives you the mean of the data. For example, with Data Set I: \(5, -2, 6, 14, -3, 0, 1, 4, 3, 2, 5\), the mean is computed as follows:
\[ \text{Mean} = \frac{5 + (-2) + 6 + 14 + (-3) + 0 + 1 + 4 + 3 + 2 + 5}{11} = \frac{35}{11} \approx 3.18 \]
A change to the data that adds a constant number to every element, as in moving from Data Set I to Data Set II, results in the mean shifting by that constant. Similarly, subtracting a constant will decrease the mean by the same constant. This transformation impacts the mean linearly, showcasing average shifts in central tendency.

The median provides another measure of central tendency by identifying the middle value in a dataset after sorting. If the dataset contains an odd number of values, the median is the middle number. For even numbers, it’s the average of the two middle numbers. Here's how to find it for Data Set I, sorted as \([-3, -2, 0, 1, 2, 3, 4, 5, 5, 6, 14]\). Since there are 11 numbers, the 6th number is the median:
\[ \text{Median} = 3 \]
When a constant is added to or subtracted from each number in the dataset, the median also shifts by that constant while maintaining its position in a sorted list. This behavior is evident when Data Set I is transformed into Data Set II, where three is added to every element, thereby increasing the median correspondingly. In Data Set III, by subtracting six, the median also decreases by that same amount, verifying a principle of consistent central position alteration when consistent changes are made across the dataset.

The mode of a dataset refers to the most frequently occurring value within the set. Calculating the mode requires examining which number appears most often. In Data Set I, for instance, the number 5 appears twice, more than any other number, indicating:
\[ \text{Mode} = 5 \]
Unlike mean and median, the mode may not shift uniformly with adding or subtracting constants because it relies on frequency. The mode can remain the same, or it can change entirely based on how frequently the numbers appear in the modified dataset, as seen when analyzing Data Set II and III. In Data Set II, 8 becomes the mode, while in Data Set III, -1 is observed most frequently, demonstrating variabilities in frequency, not directly tied to linear shifts, but influenced by resultant frequency counts and ranges of transformed values in the set.