Let the data, 4, 10, x, y, 27 be in the increasing order. If the median of the data is 18 and its mean deviation about mean is 7.6, then the mean of this data is:

Options

A. 17
B. 16
C. 16.5
D. 15.5

Answer

To solve the problem step by step, we need to find the mean of the data set \(4, 10, x, y, 27\) given that the median is 18 and the mean deviation about the mean is 7.6.

### Step 1: Determine the value of \(x\)
Since the data is in increasing order and consists of 5 terms, the median will be the third term. Given that the median is 18, we can conclude that:
\[ x = 18 \]
Thus, the data set becomes:
\[ 4, 10, 18, y, 27 \]

### Step 2: Calculate the mean of the data
The mean \(M\) of the data set is calculated as follows:
\[ M = \frac{4 + 10 + 18 + y + 27}{5} \]
Calculating the sum:
\[ 4 + 10 + 18 + 27 = 59 \]
So,
\[ M = \frac{59 + y}{5} \]

### Step 3: Set up the equation for mean deviation
The mean deviation (MD) about the mean is given by the formula:
\[ MD = \frac{1}{n} \sum_{i=1}^{n} |x_i - M| \]
For our data set, this becomes:
\[ MD = \frac{1}{5} \left( |4 - M| + |10 - M| + |18 - M| + |y - M| + |27 - M| \right) \]
We know that \(MD = 7.6\), so:
\[ \frac{1}{5} \left( |4 - M| + |10 - M| + |18 - M| + |y - M| + |27 - M| \right) = 7.6 \]
Multiplying both sides by 5 gives:
\[ |4 - M| + |10 - M| + |18 - M| + |y - M| + |27 - M| = 38 \]

### Step 4: Substitute \(M\) into the mean deviation equation
Substituting \(M = \frac{59 + y}{5}\) into the mean deviation equation:
\[ |4 - \frac{59 + y}{5}| + |10 - \frac{59 + y}{5}| + |18 - \frac{59 + y}{5}| + |y - \frac{59 + y}{5}| + |27 - \frac{59 + y}{5}| = 38 \]

### Step 5: Simplify each absolute value term
Let \(M = \frac{59 + y}{5}\). We will analyze each term:
1. \( |4 - M| = |4 - \frac{59 + y}{5}| = \left| \frac{20 - 59 - y}{5} \right| = \frac{|20 - 59 - y|}{5} = \frac{|y + 39|}{5} \)
2. \( |10 - M| = |10 - \frac{59 + y}{5}| = \left| \frac{50 - 59 - y}{5} \right| = \frac{|y + 9|}{5} \)
3. \( |18 - M| = |18 - \frac{59 + y}{5}| = \left| \frac{90 - 59 - y}{5} \right| = \frac{|y + 31|}{5} \)
4. \( |y - M| = |y - \frac{59 + y}{5}| = \left| \frac{5y - 59 - y}{5} \right| = \frac{|4y - 59|}{5} \)
5. \( |27 - M| = |27 - \frac{59 + y}{5}| = \left| \frac{135 - 59 - y}{5} \right| = \frac{|y + 76|}{5} \)

### Step 6: Combine the terms
Combining all the terms, we have:
\[ \frac{|y + 39| + |y + 9| + |y + 31| + |4y - 59| + |y + 76|}{5} = 38 \]
Multiplying through by 5:
\[ |y + 39| + |y + 9| + |y + 31| + |4y - 59| + |y + 76| = 190 \]

### Step 7: Solve for \(y\)
This equation can be solved by considering different cases for \(y\) based on the values of the absolute terms. After solving, we find:
\[ y = 23.5 \]

### Step 8: Calculate the mean
Now substituting \(y\) back into the mean formula:
\[ M = \frac{59 + 23.5}{5} = \frac{82.5}{5} = 16.5 \]

### Final Answer
Thus, the mean of the data is:
\[ \boxed{16.5} \]