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NCERT Exemplar · Q45

Q.The mean deviation of the data is _______ when measured from the median.

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The sum of absolute deviations ∑∣xi−A∣\sum\lvert x_i-A\rvert is smallest when AA is the median, so the mean deviation is minimum when measured from the median.

The property

For any data set, the mean deviation about a point AA is

M.D.(A)=1n∑i=1n∣xi−A∣.\text{M.D.}(A)=\frac{1}{n}\sum_{i=1}^{n}\lvert x_i-A\rvert.

As AA changes, this value changes. A standard result of statistics is that ∑∣xi−A∣\sum\lvert x_i-A\rvert attains its least value exactly when AA equals the median of the data.

Why the median minimizes it …

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