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Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?

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In summary, the sum of deviations from median is always minimum compared to deviations from mean, mode, or any other observation in a sample. This is because the median minimizes the sum of deviations as a function of |x_i - a|, while the mean minimizes it as a function of (x_i - a)^2. There is no known minimizing property for the mode.

- #1

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Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?

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- #2

Science Advisor

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gianeshwar said:

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.

- #3

Homework Helper

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[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.

- #4

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- #5

Homework Helper

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OK - I didn't know of any useful minimization property for the mode.

- #6

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Thank You Friends!

The sum of deviations from median is a statistical measure that calculates the total distance of a set of data points from their median value. It is used to understand the overall spread or variability of a dataset.

The sum of deviations from median is calculated by finding the difference between each data point and the median, then adding all of these differences together. The resulting value represents the total deviation of the data from the median.

A high sum of deviations from median indicates that the data points are spread out from the median, suggesting a high variability or dispersion within the dataset. This could also indicate the presence of outliers or extreme values in the data.

The sum of deviations from median is often used as a measure of variability, along with other measures such as standard deviation and range. It can also be used to identify outliers or extreme values in a dataset.

Yes, the sum of deviations from median can be negative. This would occur when the data points are mostly below the median, resulting in negative differences from the median. However, when the data points are evenly distributed around the median, the sum of deviations from median will be zero.

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