The median of 27 observations of a variable is 18 Three more observations are made and the values of these observations are 16. 18 and 50. What is the median of these 30 observation

Options

A. 18
B. 19
C. 25.5
D. Cannot be determined.

Answer

To find the median of the 30 observations after adding three new values (16, 18, and 50) to the original 27 observations, we can follow these steps:

### Step 1: Understand the given information
We know that the median of the original 27 observations is 18. This means that when the 27 observations are arranged in ascending order, the 14th observation (the middle value) is 18.

### Step 2: Add the new observations
The three new observations are 16, 18, and 50. We need to incorporate these into the original set of 27 observations.

### Step 3: Arrange the observations
To find the new median, we need to arrange all 30 observations in ascending order. Since we don't know the exact values of the original 27 observations, we can denote them as follows:
- Let’s assume the 27 observations are arranged in ascending order, and we know that the 14th observation is 18.

Now, we add the new observations:
- The new observations are 16, 18, and 50.

### Step 4: Determine the position of the new observations
When we add 16, 18, and 50 to the original observations, we need to find where these values fit in the ordered list:
- 16 will be less than or equal to some of the original observations.
- 18 will be equal to the 14th observation.
- 50 will be greater than most of the original observations.

### Step 5: Count the total observations
Now we have a total of 30 observations. To find the median, we need to find the middle value(s):
- Since there are 30 observations (an even number), the median will be the average of the 15th and 16th observations.

### Step 6: Analyze the positions
- The 14th observation is 18.
- The 15th observation will depend on the values of the original observations. However, since we know that 16 is added and is less than 18, it is likely that the 15th observation will be either 18 or the next higher value in the original set.

### Step 7: Determine the median
If we assume that the 14th observation is 18 and the next few observations are also around 18, we can conclude:
- The 15th observation will likely still be 18 or slightly higher.
- The 16th observation will also likely be 18 or slightly higher.

Thus, both the 15th and 16th observations are likely to be 18.

### Step 8: Calculate the median
The median of the 30 observations will be the average of the 15th and 16th observations:
\[ \text{Median} = \frac{\text{15th observation} + \text{16th observation}}{2} = \frac{18 + 18}{2} = 18 \]

### Final Answer
The median of the 30 observations is **18**.