Question

Consider the dataset: 4, 6, 8, 3k, 13, 17, 21, 24, known to be in ascending order. If the mean deviation about the median is exactly 6, compute the median.

• A. 10
• B. 12
• C. 11
• D. 13

Answer 1

Answer: (C)

11

Answer 2

The dataset is given as 4, 6, 8, 3k, 13, 17, 21, 24, arranged in ascending order.
The number of observations is n = 8 (an even number).
The median (M) is the average of the (n/2)-th and (n/2 + 1)-th observations.
Here, the 4th observation is 3k and the 5th observation is 13.
So, the median M = (3k + 13) / 2.

Since the data is in ascending order, we must have 8 ≤ 3k ≤ 13.
This implies 8/3 ≤ k ≤ 13/3, which is approximately 2.67 ≤ k ≤ 4.33.

The mean deviation about the median (MD) is given by the formula:
MD = (1/n) * Σ |xi - M|
We are given MD = 6 and n = 8.
So, Σ |xi - M| = n * MD = 8 * 6 = 48.

The sum of absolute deviations is:
Σ |xi - M| = |4 - M| + |6 - M| + |8 - M| + |3k - M| + |13 - M| + |17 - M| + |21 - M| + |24 - M|

Since the data is in ascending order and M is the median, the first four observations (4, 6, 8, 3k) are less than or equal to M, and the last four observations (13, 17, 21, 24) are greater than or equal to M.
Thus, for xi ≤ M, |xi - M| = M - xi.
And for xi ≥ M, |xi - M| = xi - M.

Σ |xi - M| = (M - 4) + (M - 6) + (M - 8) + (M - 3k) + (13 - M) + (17 - M) + (21 - M) + (24 - M)
Σ |xi - M| = (M + M + M + M - M - M - M - M) + (-4 - 6 - 8 - 3k + 13 + 17 + 21 + 24)
Σ |xi - M| = (4M - 4M) + (-18 - 3k + 75)
Σ |xi - M| = 57 - 3k

We know that Σ |xi - M| = 48.
So, 57 - 3k = 48.
3k = 57 - 48
3k = 9
k = 3.

Let's check if k=3 is consistent with the ascending order condition: 2.67 ≤ 3 ≤ 4.33. Yes, it is.
With k=3, the 4th observation is 3k = 3 * 3 = 9.
The dataset is 4, 6, 8, 9, 13, 17, 21, 24, which is indeed in ascending order.

Now, we can compute the median using k=3:
M = (3k + 13) / 2
M = (3 * 3 + 13) / 2
M = (9 + 13) / 2
M = 22 / 2
M = 11.

Let's verify the mean deviation with M=11 and k=3 (data: 4, 6, 8, 9, 13, 17, 21, 24):
|4 - 11| = 7
|6 - 11| = 5
|8 - 11| = 3
|9 - 11| = 2
|13 - 11| = 2
|17 - 11| = 6
|21 - 11| = 10
|24 - 11| = 13
Sum of absolute deviations = 7 + 5 + 3 + 2 + 2 + 6 + 10 + 13 = 48.
Mean deviation = Σ |xi - M| / n = 48 / 8 = 6.
This matches the given information.

The median is 11.