How are the data sets the same? How do they differ?
A. The three data sets have the same range and mean but have different standard deviations
B. The three data sets have the same standard deviations but have different ranges °
C. The three data sets have the same range but have different means
D. The three data sets have the same mean but have different standard deviations
The Correct Answer and Explanation is :
The correct answer is B. The three data sets have the same standard deviations but have different ranges.
Explanation:
In statistics, understanding how data sets compare involves examining measures of central tendency and variability. The key measures to consider are:
- Mean: The average of all data points.
- Range: The difference between the maximum and minimum values in the data set.
- Standard Deviation: A measure of the amount of variation or dispersion in a set of values.
Analyzing the Options:
- Option A: “The three data sets have the same range and mean but have different standard deviations.”
- This suggests that while the data sets are centered around the same value (mean) and spread over the same interval (range), the spread of individual data points (standard deviation) differs.
- However, if the range and mean are identical, the standard deviation would also be the same, as it depends on the spread of data points.
- Therefore, this option is incorrect.
- Option B: “The three data sets have the same standard deviations but have different ranges.”
- This indicates that the variability of data points around the mean is consistent across all data sets, but the overall spread (range) differs.
- This scenario is plausible because standard deviation measures the average distance of data points from the mean, while range measures the total spread from the minimum to the maximum value.
- Therefore, this option is correct.
- Option C: “The three data sets have the same range but have different means.”
- This suggests that the data sets cover the same interval but are centered at different points.
- While this is possible, it doesn’t directly address the relationship between standard deviation and range.
- Therefore, this option is not the best choice.
- Option D: “The three data sets have the same mean but have different standard deviations.”
- This indicates that the data sets are centered around the same value but have different spreads.
- However, this doesn’t address the range of the data sets.
- Therefore, this option is not the best choice.
Conclusion:
The most accurate description is Option B, where the data sets have the same standard deviations but different ranges. This means that while the variability of data points around the mean is consistent, the total spread of the data differs across the sets.