If two data sets have the same range: a) the distances from the smallest to largest observations in both sets will be the same b) the smallest and largest observations are the same in both sets c) both sets will have the same mean d) both sets will have the same interquartile range e) none of these
The correct answer and explanation is:
Correct Answer: a) the distances from the smallest to largest observations in both sets will be the same
Explanation:
The range of a data set is defined as the difference between the largest and smallest values in that set. Mathematically, it is calculated as:
Range = Maximum value − Minimum value
If two data sets have the same range, it means that the numerical difference between the smallest and the largest observations in each data set is equal. This does not necessarily mean that the actual values of the smallest and largest observations are the same; only that the distance between them is equal.
For example, consider the following two data sets:
- Set A: {2, 4, 6, 8, 10} → Range = 10 − 2 = 8
- Set B: {5, 7, 9, 11, 13} → Range = 13 − 5 = 8
Both sets have a range of 8, but their smallest and largest values are different. Therefore, option b is incorrect.
Having the same range does not imply the same mean. The average of Set A is 6, while the average of Set B is 9. So option c is also incorrect.
Likewise, the interquartile range (IQR), which measures the spread of the middle 50 percent of the data, can be different even if the overall range is the same. So option d is not necessarily true.
Therefore, the only statement that must be true when two data sets have the same range is option a: the distances from the smallest to largest observations in both sets will be the same. It describes what range measures without assuming anything else about the distribution of the data.