Is the Sum of Deviations from the Median Always the Smallest?

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Discussion Overview

The discussion centers around whether the sum of deviations from the median is always the smallest compared to deviations from the mean, mode, or other statistics. Participants explore the properties of these measures in relation to minimizing sums of deviations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that the median minimizes the sum of absolute deviations from a set of values.
  • Others clarify that the mean minimizes the sum of squared deviations from the values.
  • A participant notes a lack of known minimizing properties for the mode.
  • There is a reference to Wikipedia for additional information regarding these properties.

Areas of Agreement / Disagreement

Participants express differing views on the minimizing properties of the median, mean, and mode, indicating that multiple competing views remain without a consensus.

Contextual Notes

Some assumptions regarding the definitions of the statistics and the nature of the data being discussed may be implicit but are not explicitly stated.

gianeshwar
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Dear Friends!
Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?
 
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gianeshwar said:
Dear Friends!
Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?

I assume you're talking about the sum of the deviations of the values in a sample from a statistic computed from a sample. The sum of the deviations of sample values from the sample mean is zero.
 
A median minimizes this sum (in a sample) as a function of [itex]a[/itex]
[tex] \sum_{i=1}^n |x_i - a|[/tex]

The mean (sample average if you will) minimizes this sum as a function of [itex]a[/itex]
[tex] \sum_{i=1}^n \left(x_i - a\right)^2[/tex]

I don't know of any minimizing property for the mode.
 
OK - I didn't know of any useful minimization property for the mode.
 
Thank You Friends!
 

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