Following are the observations of a data sample in ascending order. If the mean of the sample is 18.5 and the mode is 15.5, find the missing data items. 12.5 13.8 15.5 x 19.2 20.4 y 23.5 24.6

Answer

To solve the problem step-by-step, we will find the missing data items \( x \) and \( y \) based on the given mean and mode.

### Step 1: Understand the Data
The given data in ascending order is:
\[ 12.5, 13.8, 15.5, x, 19.2, 20.4, y, 23.5, 24.6 \]

We know:
- Mean = 18.5
- Mode = 15.5

### Step 2: Determine the Value of \( x \) Using Mode
The mode is the value that appears most frequently in a data set. Since 15.5 is the mode and it appears only once in the current data, it must appear again. The only place it can appear again, while maintaining the ascending order, is where \( x \) is located.

Thus, we conclude:
\[ x = 15.5 \]

### Step 3: Update the Data Set
Now we can update our data set with the value of \( x \):
\[ 12.5, 13.8, 15.5, 15.5, 19.2, 20.4, y, 23.5, 24.6 \]

### Step 4: Calculate the Total Number of Observations
The total number of observations is 9.

### Step 5: Use the Mean to Find \( y \)
The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Given that the mean is 18.5, we can set up the equation:
\[ 18.5 = \frac{\text{Sum of all observations}}{9} \]
Thus, the sum of all observations can be calculated as:
\[ \text{Sum of all observations} = 18.5 \times 9 = 166.5 \]

### Step 6: Calculate the Current Sum of Known Values
Now we will calculate the sum of the known values:
\[ 12.5 + 13.8 + 15.5 + 15.5 + 19.2 + 20.4 + y + 23.5 + 24.6 \]
Calculating the sum of the known values (excluding \( y \)):
\[ 12.5 + 13.8 + 15.5 + 15.5 + 19.2 + 20.4 + 23.5 + 24.6 = 147.5 \]

### Step 7: Set Up the Equation to Solve for \( y \)
Now we can set up the equation:
\[ 147.5 + y = 166.5 \]
To find \( y \), we subtract 147.5 from both sides:
\[ y = 166.5 - 147.5 = 19.0 \]

### Step 8: Final Values
Thus, the missing data items are:
- \( x = 15.5 \)
- \( y = 19.0 \)

### Summary of the Solution
The values of \( x \) and \( y \) are:
- \( x = 15.5 \)
- \( y = 19.0 \)