If the mean of 26 , 19 , 15 , 24 , and x is x , then find the median of the data .

Options

A. 23
B. 22
C. 20
D. 21

Explanation

To solve the problem step by step, we need to find the value of \( x \) first, and then calculate the median of the data set.

### Step 1: Set up the equation for the mean
The mean of the numbers \( 26, 19, 15, 24, \) and \( x \) is given to be \( x \). The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the sum of the observations is \( 26 + 19 + 15 + 24 + x \) and the number of observations is \( 5 \). Therefore, we can write:
\[ x = \frac{26 + 19 + 15 + 24 + x}{5} \]

### Step 2: Simplify the equation
Now, let's calculate the sum of the known numbers:
\[ 26 + 19 = 45 \]
\[ 45 + 15 = 60 \]
\[ 60 + 24 = 84 \]
So, we have:
\[ x = \frac{84 + x}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
To eliminate the fraction, we can cross-multiply:
\[ 5x = 84 + x \]

### Step 4: Solve for \( x \)
Now, we can isolate \( x \) by subtracting \( x \) from both sides:
\[ 5x - x = 84 \]
\[ 4x = 84 \]
Now, divide both sides by \( 4 \):
\[ x = \frac{84}{4} = 21 \]

### Step 5: List the observations
Now that we have found \( x = 21 \), we can list all the observations:
\[ 15, 19, 21, 24, 26 \]

### Step 6: Arrange the observations in increasing order
The observations in increasing order are:
\[ 15, 19, 21, 24, 26 \]

### Step 7: Find the median
To find the median of a data set with an odd number of observations, we use the formula:
\[ \text{Median} = \text{The } \left(\frac{n + 1}{2}\right)^{th} \text{ term} \]
Here, \( n = 5 \):
\[ \text{Median} = \left(\frac{5 + 1}{2}\right)^{th} \text{ term} = 3^{rd} \text{ term} \]
The 3rd term in our ordered list is \( 21 \).

### Final Answer
Thus, the median of the data is:
\[ \text{Median} = 21 \]