Connection between mean and median

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Discussion Overview

The discussion centers on the relationship between the mean and median of a dataset, particularly in the context of increasing the number of data points. Participants explore whether the difference between the mean and median decreases as the sample size increases, with a focus on the implications for symmetric and asymmetric distributions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant posits that with a larger dataset, the probability that the mean and median are close (within 0.1 or 0.2) increases, suggesting that for very large datasets, this difference approaches 0.
  • Another participant argues that the mean and median are only the same for symmetric distributions, providing an example of a small dataset to illustrate this point.
  • Some participants contend that the relationship between mean and median approaching similarity applies specifically to symmetric distributions, questioning the relevance of examples with small numbers of data points.
  • There is a challenge regarding the interpretation of examples, with one participant noting that a proposed distribution with three values does not adequately represent the question of large datasets.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the mean and median are similar, with some emphasizing the role of distribution symmetry. The discussion remains unresolved regarding the implications of increasing dataset size on the relationship between mean and median.

Contextual Notes

Limitations include the dependence on the type of distribution and the size of the dataset, as well as the potential for misunderstanding examples used in the discussion.

dextercioby
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TL;DR
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.
 
Last edited:
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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