Let the median and the mean deviation about the median of 7 observations 170, 125, 230, 190, 210, a, b be 170 and respectively. Then the mean deviation about the mean of these 7 observations is

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Solution

Q.
Let the median and the mean deviation about the median of 7 observations 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is: [2024]

1 32

2 30

3 28

4 31

Ans.
(2)

The ascending order of 7 observations are 125, a, b, 170, 190, 210, 230

$M.D. (Med) = \frac{1}{7} \sum_{i = 1}^{7} | x_{i} - Med | = \frac{205}{7}$

$\Rightarrow | 125 - 170 | + | a - 170 | + | b - 170 | + | 170 - 170 | + | 190 - 170 | + | 210 - 170 | + | 230 - 170 | = 205$

$[∵ Median = 170]$

$\Rightarrow a + b = 300$ $. . . (i)$

$∴ Mean = \frac{125 + 300 + 170 + 190 + 210 + 230}{7} = \frac{1225}{7} = 175$ (By (i))

$∴ M.D. (Mean) = \frac{1}{7} \sum_{i = 1}^{7} | x_{i} - Mean |$ $= \frac{1}{7} (50 + 350 - 300 + 5 + 15 + 35 + 55)$

$= \frac{210}{7} = 30$

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