Final answer:
The standard deviation (SD) is a measure of how spread out the numbers in a data set are. It is independent of the mean's value and therefore, if the mean becomes smaller but the distances between the data points stay the same, the standard deviation remains the same.
Step-by-step explanation:
When considering the relationship between the standard deviation (SD) and the mean (µ) of a distribution, changing the mean does not affect the standard deviation. The standard deviation is a measure of the spread or variability of the data. It is determined by how far each data point is from the mean. If the mean becomes smaller, but the relative distances between the data points remain unchanged, then the standard deviation remains the same (Option A). Thus, the standard deviation is independent of the mean's value; it is only affected by the differences between individual data points and the mean.
For example, if the dataset is {2, 4, 6}, and the mean is 4, reducing all values by 1 to get the dataset {1, 3, 5} will result in a smaller mean, which is 3. However, the spread of the numbers around the mean is unchanged, and so the standard deviation remains the same.
It is also important to note that the standard deviation is always positive or zero. A standard deviation of zero indicates that all data points are equal, signifying no variability within the dataset (Option C under point 15).