If two groups of numbers have the same mean, then a. their standard deviations must also be equal. b. their medians must also be equal. c. their modes must also be equal. d. other measures of location need not be the same
The Correct Answer and Explanation is:
Correct answer: d. other measures of location need not be the same
Two groups of numbers can share the same mean while differing in many other statistical aspects. The mean, or average, is simply the sum of the values divided by the number of values. It is a measure of central tendency, but it does not fully describe the distribution or spread of the data. Therefore, knowing that two datasets have the same mean does not guarantee equality in other statistics such as standard deviation, median, or mode.
Standard deviation measures the spread or dispersion of a dataset. Two datasets can have identical means but very different spreads. For example, the sets {4, 6} and {0, 10} both have a mean of 5, but the second set has a much larger spread, thus a larger standard deviation.
The median, another measure of central tendency, represents the middle value when a dataset is ordered. It is resistant to extreme values. A dataset like {1, 5, 9} has a mean and median of 5, while {2, 5, 8} also has a mean of 5 but a slightly different data distribution, which can influence the median in other cases.
The mode refers to the most frequently occurring value in a dataset. This measure is particularly sensitive to repetition. For example, in {3, 5, 5, 7}, the mode is 5, but in {3, 4, 6, 7}, there is no mode. Yet both datasets could have the same mean.
Because each statistical measure captures different characteristics of a dataset, having one measure (like the mean) be equal does not imply that the others are also equal. Thus, option d correctly captures this concept: other measures of location or spread need not be the same.
