A set of numbers is given as 8, 9, x, (x,3), 15. The mean of these numbers is 11. Find the value of x and the standard deviation.

The numbers are $8, 9, x, x+3, 15$.
The mean ($\bar{x}$) is 11.
The number of terms ($N$) is 5.

The mean is calculated as the sum of all terms divided by the number of terms:

Multiply both sides by 5:

Subtract 35 from both sides:

Divide by 2:

The value of $x$ is $\boxed{\text{10}}$.

Substitute $x=10$ into the list of numbers:

The data set is:

The mean ($\bar{x}$) is 11.

The variance is the average of the squared differences from the mean. The formula for population variance is $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}$.
First, calculate the deviations from the mean $(x_i - \bar{x})$ and their squares $(x_i - \bar{x})^2$:

• For $x_1 = 8$: $(8 - 11)^2 = (-3)^2 = 9$
• For $x_2 = 9$: $(9 - 1