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 :

Question

If the mean deviation about the median of the numbers \[ k,\,2k,\,3k,\,\ldots,\,1000k \] is \(500\), then \(k^2\) is equal to

Answer 1

With a median of 170, the sorted observations are: 125, a, b, 170, 190, 210, 230.

The mean deviation from the median is calculated as:

\[ \frac{0 + |45| + |60| + |20| + |40| + |170 - a| + |170 - b|}{7} = \frac{205}{7} \]

This yields:

\(|170 - a| + |170 - b| = 300 \implies a + b = 300\)

The mean of the observations is then:

\[ \text{Mean} = \frac{125 + a + b + 170 + 190 + 210 + 230}{7} = \frac{125 + 300 + 170 + 190 + 210 + 230}{7} = 175 \]

The mean deviation from the mean is:

\[ \frac{|125 - 175| + |a - 175| + |b - 175| + |170 - 175| + |190 - 175| + |210 - 175| + |230 - 175|}{7} \]

Simplifying the expression:

\[ \frac{50 + |a - 175| + |b - 175| + 5 + 15 + 35 + 55}{7} = 30 \]

The Correct Option is C