A data set has `13` distinct observation with median `18`. If one more number is added to this data set. The median will remain unchanged. Find the number added.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Given Information
We have a dataset with 13 distinct observations and a median of 18.

**Hint:** Recall that the median is the middle value when the numbers are arranged in order.

### Step 2: Determine the Position of the Median
Since there are 13 observations (an odd number), the median is the value at the position \((n + 1) / 2\), where \(n\) is the number of observations.

\[ \text{Position of median} = \frac{13 + 1}{2} = \frac{14}{2} = 7 \]

**Hint:** The median is the 7th term in a sorted list of 13 numbers.

### Step 3: Identify the Value of the Median
According to the problem, the 7th term (median) is given as 18.

**Hint:** This means that when the numbers are sorted, the 7th number is 18.

### Step 4: Add One More Number to the Dataset
Now, we add one more number to the dataset, making it 14 observations. We need to find out what this number can be such that the median remains unchanged at 18.

**Hint:** When you add a number, the total count of numbers becomes even, which changes how the median is calculated.

### Step 5: Calculate the New Median Position
With 14 observations, the median is now the average of the 7th and 8th terms.

**Hint:** For an even number of observations, the median is calculated as the average of the two middle numbers.

### Step 6: Set Up the Equation for the New Median
Let’s denote the 7th term as \(x_7\) and the 8th term as \(x_8\). We know:

\[ \text{New Median} = \frac{x_7 + x_8}{2} \]

Since the median must remain 18, we have:

\[ \frac{x_7 + x_8}{2} = 18 \]

Multiplying both sides by 2 gives:

\[ x_7 + x_8 = 36 \]

**Hint:** You already know \(x_7 = 18\) (the median).

### Step 7: Solve for the 8th Term
Substituting \(x_7 = 18\) into the equation:

\[ 18 + x_8 = 36 \]

Subtracting 18 from both sides:

\[ x_8 = 36 - 18 = 18 \]

**Hint:** This means the 8th term must also be 18.

### Step 8: Conclusion
Since the dataset originally had 13 distinct numbers, and now we have added the number 18, we conclude that the number added to the dataset is:

\[ \text{Number added} = 18 \]

**Final Answer:** The number added is 18.