Question

A researcher computed the mean, the median, and the standard deviation for a set of performance scores. If 5 were to be added to each score, which of these three statistics would change?

• A. The mean only
• B. The median only
• C. The standard deviation only
• D. The mean and the median
• E. The mean and the standard deviation

Hint:

Remember these simple rules for data transformation: - Adding/subtracting a constant changes measures of center (mean, median) but not measures of spread (range, standard deviation). - Multiplying/dividing by a constant changes both measures of center and measures of spread.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the understanding of how basic statistical measures (mean, median, standard deviation) are affected when a constant value is added to every data point in a set.
Step 2: Detailed Explanation:
Let's analyze the effect of adding 5 to each score on each statistic:
Mean: The mean is the sum of all scores divided by the number of scores. If we add 5 to each of the \(n\) scores, the total sum increases by \(5n\). The new mean will be \((\text{Old Sum} + 5n) / n = (\text{Old Sum} / n) + (5n / n) = \text{Old Mean} + 5\). Therefore, the mean changes.
Median: The median is the middle value of a sorted dataset. When 5 is added to every score, the relative order of the scores does not change. The score that was in the middle will still be in the middle, but its value will have increased by 5. For example, if the scores are {10, 20, 30}, the median is 20. If we add 5, the scores become {15, 25, 35}, and the new median is 25. Therefore, the median changes.
Standard Deviation: The standard deviation measures the spread or dispersion of the data points from the mean. When we add a constant to every score, the entire dataset shifts along the number line, but the distances between the scores remain the same. The mean also shifts by the same constant, so the distance of each score from the mean \((x_i - \text{mean})\) does not change. Since the standard deviation is calculated based on these distances, it remains unchanged. Therefore, the standard deviation does not change.
Step 3: Final Answer:
Both the mean and the median would change, while the standard deviation would not. Thus, the correct option is the mean and the median.