If two groups of numbers have the same mean, then their modes must also be equal. b. Medians must also be equal. None of the other answers are correct: Standard deviations must also be equal.
The Correct Answer and Explanation is:
Correct Answer: d. Other measures of location need not be the same
Explanation:
Two groups of numbers can share the same arithmetic mean and still differ significantly in other statistical characteristics such as the mode, median, and standard deviation. The mean represents the average value of a dataset, calculated by summing all values and dividing by the number of values. However, this single measure does not fully describe the distribution of data.
The mode is the most frequently occurring value in a dataset. A group can be unimodal, bimodal, or even have no mode. Two datasets might have the same average but different most frequent values, leading to distinct modes.
The median is the middle value when data are arranged in order. If one dataset is skewed or contains outliers, its median may differ significantly from another dataset with the same mean but a more symmetrical distribution.
The standard deviation measures how spread out the values are around the mean. A dataset with values tightly clustered around the mean will have a smaller standard deviation, whereas a dataset with values far from the mean will have a larger one. Identical means do not imply equal variability.
To illustrate, consider the following datasets:
- Group A: {2, 4, 6}
- Group B: {0, 6, 6}
Both groups have a mean of 4. However:
- Group A’s mode does not exist; Group B’s mode is 6.
- Group A’s median is 4; Group B’s median is 6.
- Group A’s standard deviation is approximately 2; Group B’s is larger due to the more extreme values.
This example confirms that equal means do not necessitate equal modes, medians, or standard deviations. Therefore, none of the other options must be true solely because the means are equal.
