True or False: The mean, median, and mode are always the same for any given dataset.

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BlackTom AI · Accepted Answer

Question restatement: True or False: The mean, median, and mode are always the same for any given dataset. Options: 
- True
- False

Option 1 (True): This would claim that in every dataset the mean, median, and mode coincide. While there are special cases where they match (for example, a perfectly symmetric, unimodal dataset or a dataset with all identical values), this is not guaranteed for all datasets. A clear counterexample shows the failure of this statement: consider the dataset [1, 2, 3, 4, 100]. The mean is (1+2+3+4+100)/5 = 110/5 = 22, the median is the middle value 3, and the mode is not uniquely defined (there is no value that occurs more than once). Since the three measures do not equal each other in general, this option is incorrect.

Option 2 (False): This option asserts that the equality of mean, median, and mode does not hold for all datasets. The explanation above demonstrates a scenario where they differ, confirming that the statement is not universally true. In statistics, the mean, median, and mode can be the same for certain datasets (e.g., symmetric, unimodal distributions or datasets with all identical values), but they are not guaranteed to be the same in every case. Additionally, the mode may be undefined or multiple values may share the highest frequency, further illustrating why the three measures need not align in all situations. Therefore, this option correctly captures the general truth about these three measures.

Summary of misconceptions addressed: - Assuming a dataset’s symmetry guarantees identical mean, median, and mode is incorrect because symmetry alone doesn’t ensure identical central tendency measures in all cases. - Believing the mode always exists ignores scenarios with all distinct values or multimodal distributions. - Thinking only about one example is insufficient; a robust answer requires a general understanding of why these three statistics can diverge across different datasets.

STP420 - Probability and Statistics Pre-Test STP420 - Probability and Statistics Pre-Test

True or False: The mean, median, and mode are always the same for any given dataset.

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