I Connection between mean and median

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The discussion centers on the relationship between the mean and median in datasets of varying sizes. It is noted that the mean and median will only be equal in distributions that are symmetric around the mean. As the dataset size increases, particularly to very large numbers, the difference between the mean and median is likely to decrease, approaching zero for symmetric distributions. However, for non-symmetric distributions, this convergence does not necessarily hold true. The example provided illustrates that with a small dataset, the mean and median can differ significantly, highlighting the importance of distribution shape.
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See title and problem below.
I have 100 random real (even rational with only one decimal, like average temperatures of months at a particular weather station) numbers. With them I compute the arithmetical mean and the median. It is a (very) small probability they are the same number within let's say 0,1 or 0,2.

Question. If I let the number of items increase (let us say 1000 instead of 100), is it more probable that the difference between the mean and the median decreases? It is true that for a very, very large number of numbers this difference is arbitrarily close to 0, FAPP is 0?
 
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Mean and median will only be the same for a distribution symmetric around the mean. Example: (0,1,10) has a median of 1 and a mean of 11/3.
 
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I think that only applies to symmetric distributions, that the mean and median will approach the same value for very very large data sample.

mathman said:
Mean and median will only be the same for a distribution symmetric around the mean. Example: (0,1,10) has a median of 1 and a mean of 11/3.
The question was what will happen if you have very very large dataset. Your example has three numbers.
 
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malawi_glenn said:
I think that only applies to symmetric distributions, that the mean and median will approach the same value for very very large data sample.The question was what will happen if you have very very large dataset. Your example has three numbers.

No, they're suggesting a distribution that takes three values with equal probability
 
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