For what value of x, the mode of the following dataL 13,24,13,27,17,16,17,x,22,21,13 is 17?

Answer

To find the value of \( x \) such that the mode of the data set \( 13, 24, 13, 27, 17, 16, 17, x, 22, 21, 13 \) is 17, we will follow these steps:

### Step 1: Identify the current frequencies of the data
First, we need to count how many times each number appears in the data set without considering \( x \):
- \( 13 \): appears 3 times
- \( 24 \): appears 1 time
- \( 27 \): appears 1 time
- \( 17 \): appears 2 times
- \( 16 \): appears 1 time
- \( 22 \): appears 1 time
- \( 21 \): appears 1 time

### Step 2: Determine the condition for mode
The mode is the number that appears most frequently in the data set. For the mode to be 17, it must appear more frequently than any other number. Currently, \( 13 \) appears 3 times, and \( 17 \) appears 2 times.

### Step 3: Set up the equation for \( x \)
To ensure that the mode is 17, the frequency of 17 must be greater than the frequency of 13. Since 13 appears 3 times, we need 17 to appear at least 4 times.

Currently, 17 appears 2 times, and if we set \( x = 17 \), it will appear 3 times. Thus, we need one more occurrence of 17 to make it 4 times. Therefore, we can conclude that:

\[ \text{Frequency of } 17 = 2 + 1 = 3 \text{ (if } x = 17\text{)} \]

### Step 4: Solve for \( x \)
To make the frequency of 17 equal to 4, we set \( x = 17 \):

\[ x = 17 \]

### Conclusion
Thus, the value of \( x \) that makes the mode of the data set equal to 17 is:

\[ \boxed{17} \]