Final answer:
For various distributions, the mean and median can be the same or different, depending on whether the distribution is uniform, skewed, or symmetric. In uniform and symmetric non-skewed distributions, the mean equals the median, whereas in skewed distributions, the mean is pulled toward the tail.
Step-by-step explanation:
Understanding Distribution of Data
When analyzing data, the mean, median, and mode are essential measures that provide different insights. For a uniform distribution, data is evenly spread, and the mean and median should be at the midpoint of the range. For a dataset that is skewed right, the mean is larger than the median because the longer tail is on the right. Conversely, in a skewed left distribution, the mean is smaller than the median due to the longer tail on the left. In a symmetric distribution, if skewed at all, the mean, median, and mode should all coincide if the distribution is normal; otherwise, the two modes differ from the mean and median in a bimodal distribution.
Below are the calculations for each scenario given:
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- A - For a uniform distribution between 4 and 12, Mean: 8, Median: 8
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- B - With a right skew and most values at 10, the Mean would be >10, and Median = 10
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- C - With a left skew and most values at 10, the Mean would be <10, and Median = 10
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- D - For a symmetric distribution with values at 4 and 16, Mean: 10, Median: 10
It's important to note that in skewed distributions, the mean is pulled toward the tail of the data.