We are given the set of numbers: $11, 8, 14, 14, x$, and the mean of these numbers is 13. We are tasked with finding the value of $x$.

The mean of a set of numbers is calculated as:

$\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Total number of values}}$

In this case, the mean is 13, and there are 5 numbers. Therefore, we can write the equation as:

$13 = \frac{11 + 8 + 14 + 14 + x}{5}$

Now, solve for $x$:

1. First, find the sum of the known numbers:

$11 + 8 + 14 + 14 = 47$

2. Substitute this sum into the equation:

$13 = \frac{47 + x}{5}$

3. Multiply both sides of the equation by 5 to eliminate the denominator:

$13 \times 5 = 47 + x$

$65 = 47 + x$

4. Finally, solve for $x$ by subtracting 47 from both sides:

$x = 65 - 47$

$x = 18$

So, the value of $x$ is 18.

Do you need further clarification or details? Here are some related questions to explore:

1. How would the mean change if the value of $x$ were different?
2. What would be the median of the set $11, 8, 14, 14, 18$?
3. How do you find the mode of a set of numbers?
4. Can you calculate the range of the set once $x$ is known?
5. How is the mean affected if one of the numbers is doubled?

Tip: To find the mean of a set, always remember to divide the total sum by the number of values in the set.