Question

Let mean and median of 9 observations 8, 13, a, 17, 21, 51, 103, b, 67 are 40 and 21 respectively where a > b. If mean deviation about median is 26 then 2a is :-

• A. 130
• B. 131
• C. 51
• D. 40

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Mean is the average of all observations. Median is the middle value of sorted data.
Mean deviation about median is defined as $\frac{1}{n}\sum |x_i - \text{median}|$.
Step 2: Key Formula or Approach:
1. Use the mean to find the sum of $a$ and $b$.
2. Use the mean deviation about median formula to set up an equation involving $a$ and $b$.
3. Account for the fact that the median is 21 and $a>b$ to solve the system.
Step 3: Detailed Explanation:
1. From Mean $= 40$:
\[ \frac{8 + 13 + 17 + 21 + 51 + 103 + 67 + a + b}{9} = 40 \]
\[ 280 + a + b = 360 \implies a + b = 80 \]
2. Given Median $= 21$. Since there are 9 observations, the 5th observation in ascending order is 21.
The known values are: 8, 13, 17, 21, 51, 67, 103.
Since 21 is the 5th term and $a>b$, $b$ must be $\le 21$ and $a \ge 21$.
3. From Mean Deviation about Median $= 26$:
\[ \frac{\sum_{i=1}^9 |x_i - 21|}{9} = 26 \implies \sum |x_i - 21| = 234 \]
Calculate deviations for known terms:
$|8-21|=13, |13-21|=8, |17-21|=4, |21-21|=0, |51-21|=30, |67-21|=46, |103-21|=82$.
Sum of known deviations $= 13 + 8 + 4 + 0 + 30 + 46 + 82 = 183$.
Remaining deviations: $|a - 21| + |b - 21| = 234 - 183 = 51$.
4. Using $a \ge 21$ and $b \le 21$:
$(a - 21) + (21 - b) = 51 \implies a - b = 51$.
5. Solve for $a$:
$(a + b) + (a - b) = 80 + 51 \implies 2a = 131$.
Step 4: Final Answer:
The value of 2a is 131.