Find Value of Parameter given Mean Median and Mean Deviation

To find the value of $2a$, we follow these steps:

Step 1: Calculate the sum of the numbers using the mean The given set consists of $9$ non-negative numbers: $21, 8, 17, a, 51, 103, b, 13, 67$. Given that the mean of these $9$ numbers is $40$: $\text{Mean} = \frac{21 + 8 + 17 + a + 51 + 103 + b + 13 + 67}{9} = 40$

$\frac{280 + a + b}{9} = 40$ $280 + a + b = 360$ $a + b = 80 \quad \text{--- (Equation 1)}$

Step 2: Use the median property to establish bounds on $a$ and $b$ Since there are $N = 9$ observations, the median is the $\left(\frac{9+1}{2}\right)^{\text{th}} = 5^{\text{th}}$ observation when arranged in ascending order. The $7$ known numbers in ascending order are: $8, 13, 17, 21, 51, 67, 103$.

Given that the median is $21$ and $a > b$, $b$ must lie to the left of or equal to $21$ ($b \le 21$), and $a$ must lie to the right of or equal to $21$ ($a \ge 21$) so that $21$ remains the $5^{\text{th}}$ observation.

Step 3: Calculate the mean deviation about the median The mean deviation about the median is given as $26$: $\text{Mean Deviation} = \frac{1}{9} \sum_{i=1}^{9} |x_i - \text{Median}| = 26$ $\sum_{i=1}^{9} |x_i - 21| = 26 \times 9 = 234$

Now, calculating the sum of absolute deviations for the $7$ known values: $|8 - 21| + |13 - 21| + |17 - 21| + |21 - 21| + |51 - 21| + |67 - 21| + |103 - 21|$ $= 13 + 8 + 4 + 0 + 30 + 46 + 82 = 183$

Including $a$ and $b$ in the sum of absolute deviations: $183 + |a - 21| + |b - 21| = 234$ $|a - 21| + |b - 21| = 51$

Since $a \ge 21$ and $b \le 21$, we have $|a - 21| = a - 21$ and $|b - 21| = 21 - b$: $(a - 21) + (21 - b) = 51$ $a - b = 51 \quad \text{--- (Equation 2)}$

Step 4: Solve for $2a$ Adding Equation (1) and Equation (2): $(a + b) + (a - b) = 80 + 51$ $2a = 131$

Thus, $2a$ is equal to 131.