Let the median and the mean deviation about the median of 7 observation 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 :
- 31
- 28
- 30
- 32
The Correct Option is C
Solution and Explanation
Given that the median is 170, the observations are arranged as:
125, a, b, 170, 190, 210, 230
The mean deviation about the median is given by:
\[ \frac{0 + |45| + |60| + |20| + |40| + |170 - a| + |170 - b|}{7} = \frac{205}{7} \]
From this, we find:
\(|170 - a| + |170 - b| = 300 \implies a + b = 300\)
Now, the mean of the observations is:
\[ \text{Mean} = \frac{125 + a + b + 170 + 190 + 210 + 230}{7} = \frac{125 + 300 + 170 + 190 + 210 + 230}{7} = 175 \]
The mean deviation about the mean is:
\[ \frac{|125 - 175| + |a - 175| + |b - 175| + |170 - 175| + |190 - 175| + |210 - 175| + |230 - 175|}{7} \]
Simplifying:
\[ \frac{50 + |a - 175| + |b - 175| + 5 + 15 + 35 + 55}{7} = 30 \]
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