Answer :
It's not possible to definitively prove that the median lies between the mean and mode for any dataset without actually looking at the data itself. However, there is a theorem that states that under certain conditions, the median will indeed fall between the mean and mode. This theorem is known as Chebyshev's inequality.
Chebyshev's inequality applies to unimodal distributions, which means there's a single most frequent value (the mode). The theorem states that a significant portion of the data (at least 1 - 1/k^2) will fall within k standard deviations of the mean, where k is any positive number greater than 1.
Here's the intuition behind the theorem:
* If the data is clustered around the mean (i.e., has low variability), the median will be close to the mean.
* As the data spreads out further from the mean (i.e., has high variability), the median will still tend to be somewhere within the range of values that are not too extreme.
In most real-world datasets, the mode is likely to be closer to the center of the data distribution than the most extreme values. Therefore, if the median is somewhere between the non-extreme values (as implied by Chebyshev's inequality), it's likely to fall between the mean and the mode.
However, it's important to remember that Chebyshev's inequality is a general principle and may not hold true for all specific cases. There can be exceptions, particularly for small datasets or highly skewed distributions.
Chebyshev's inequality applies to unimodal distributions, which means there's a single most frequent value (the mode). The theorem states that a significant portion of the data (at least 1 - 1/k^2) will fall within k standard deviations of the mean, where k is any positive number greater than 1.
Here's the intuition behind the theorem:
* If the data is clustered around the mean (i.e., has low variability), the median will be close to the mean.
* As the data spreads out further from the mean (i.e., has high variability), the median will still tend to be somewhere within the range of values that are not too extreme.
In most real-world datasets, the mode is likely to be closer to the center of the data distribution than the most extreme values. Therefore, if the median is somewhere between the non-extreme values (as implied by Chebyshev's inequality), it's likely to fall between the mean and the mode.
However, it's important to remember that Chebyshev's inequality is a general principle and may not hold true for all specific cases. There can be exceptions, particularly for small datasets or highly skewed distributions.